characteristic p of the ground field is allowed to become positive. The title is also ... The present work is an expanded version of the minicourse of 3 lectures lver ...
Canadian Society Conference Proceedings Volume 6 (1986)
AND
DUALITY
Steven 1. PREFACE. The title "Tangency and Duality over Arbitrary Fields" was given by Wallace to the article in which he pioneered the study of the similarities and differences that appear in the indicated theory, a basic topic in projective algebraic geometry, when the characteristic p of the ground field is allowed to become positive. The title is also an apt choice for the present work. However, the words "over Arbitrary Fields" were dropped for two reasons: First, the subject has ::natured to the point where it can, fairly, go without saying that p will be arbitrary. Second and more important, the similarities differences attendant to p are secondary to the geometry tself. The bulk of material is, moreover, characteristic free, and many of the special considerations required when p > 0 highl t feQtures of geometry over the complex numbers that are sometimes taken for granted. A case in point is provided by the central notion of reflexivity. In most situtations, instead of assuming p 0, t suffices to assume that the principal varieties are reflexive. A major issue then, when p > 0 , is to find useful conditions guaranteeing reflexivity. The present work is an expanded version of the minicourse of 3 lectures lver bv rhe author. The work is intended first and foremost to intrc,d"ce this lovely subject and, in particular, to announce and to introduce a fair number of recent results. An atteLlpt has been to place the results in context and in perspective, to explain their meaning and significance, and to give a feeling for their proofs. The full details of the proofs, especially if they are available elsewhere, are seldom presented. The presentation is usually expository, rarely formal. There are, however, several mathematical tidbits that are not found elsewhere. Some of these are: a fuller discussion of Wallace's construction of infinitely many plane curves with a given dual curve; a simpler and more conceptual proof of the theorem of generic order of contact, 1(101; a more traditional proof of Piene's theorem comparing the ranks of a variety with those of a general hyperplane section and those of a general ion; an account of Landman's unpublished application of Lefschetz theory to the theory of the dual variety; the application of Goldstein's theory of the second fundamental form to the stuey of toe simplicity of a general contact between two varieties one varying; a new derivation of the number, 51, of conics tangent, when p = 2 , to 5 general conics; and a new study of the limiting behavior of the tangent hyperplanes to a variety degenerating under a hOT.olography. 1980 Mathematics Subject Classification. 14-N05, 14-NIO. ISupported in part by the Americal' National Science Foundation and the Danish National Science Research Council. 19Hi A,merican '.,jathemati,al :-;OCiNy 117:{1 l\}:3ti "Ii $1.00
IG3
pN page
STEVEN L.
164
CONTENTS I.
The correspondence which gai
THE DUALITY OF PROJECTIVE VARIETIES I-I. HISTORICAL INTRODECTION 1-2. BASIC CONSIDERATIONS 1-3. NONREFLEXIVE CURVES 1-4. THE CONTACT LOCUS 1-5. THE HESSIAN 1-6. ORDINARINESS 1-7. SMOOTH VARIETIES
II.
164 167 169 172 175 178 183
quarter of the J9th cent correspondence (and in f correspondence in space traces a conic, its pola' immediately to give the
THE RANKS II-I. FUNDAMENTALS 11-2. SMOOTH VARIETIES II- 3. TOPOLOGY
III.
was developed, intensely
186 190 193
circumscribed about a concurrent.
1
COl
He derived
a hexagon is inscribed,
THE CONTACT FORMULA III-I. ABOUT CONTACTS III-2. I-PARAMETER FA.'fILIES 111-3. COMPLEMENTS 111-4. M-PARAMETER FAMILIES III-5. CONICS 111-6. OTHER CHARACTERISITIC NUMBERS III-7. SPECIALIZATION III-S. CONNECTIONS WITH SINGULARITY THEORY
197
be true.
211
Desargue's (164S) two tri
209
213 219 222
BIBLIOGRAPHY
It was soon discovered tt
200 202 205
its reciprocal, the resul Another famous
suggested that what is in does not depend on the me dual A controversy arose,
I.
Gergonne.
THE DUALITY OF PROJECTIVE VARIETIES
Poncelet right
theory of reciprocal figu because Cauchy delayed hi
1.
HISTORICAL INIRODUCTION.
(The following discussion is based on these
that a mediating conic (0'
sources: Chasles [1837J; Coxeter [1964]; Kline [1972]; Gillispe [1970-SJ.)
The
on the other hand, could
idea of passing in a natural way between the points and the lines in a plane is
The controversy was
an idea that originated in classical antiquity and was developed for over 2000
projective geometry.
years as part of the theory of conics.
calculus.
remarkable discovery. given conic
The idea is based on the following
If a variable (complete) quadrangle is inscribed in a
representing points, and
opposite sides) is a fixed point
P, then the other two diagonal points always
determine the same line
P
L
If
is outside
C, then
, then L is the tangent of C at a line L , called the polar of P If on
C
fixed line point
R.
M , then, it is not hard to see, Thus a line
p
P ; if
P
gives rise to
R, called the
Moreover, remarkably, the cross ratio of any four points L
through
P R.
on
of
rise to a point-to-point and line-to-line correspondence that preserves incidence and cross ratios and that, clearly, is of order 2.
Plucker's explanation of
c
dual triples, the geometry
M
M is equal to
In sum,
3.'l?
is
will rotate about a certain
L
symmetric relation of inci
is obviously the
is now allowed to vary along a
M gives rise to a point
the cross ratio of four corresponding lines
L
through
Thus a point
P
p
C
Fin
Then, independE
is in common use today, 0:
C so that one diagonal point (that is, the common point of two
join of the points of contact of the two tangents of
1
C gives
elements) are indistinguis
this below), a given curve
relating the coordinates c coordinates of its tangent The duality of curves reciprocity
..
holds:
TANGENCY
AND
DUALITY
165
The correspondence became the basis of a method, the method of polar reciprocation, which gained central importance in the system of geometry that
164 167 169 172
was developed, intensely, in the school of Monge in Paris during the first quarter of the 19th century.
In 1806, just after Monge had lectured on the
correspondence (and in fact on the analogous point-plane and line-line
175
correspondence in space as well), Brianchon and Livet found that, when a point
178 183
traces a conic, its polar line envelops another conic.
186
circumscribed about a conic, then the diagonals joining opposite sides are
190
concurrent.
193
197 200 202 205 209 211 213
219
222
Brianchon went on
immediately to give the theorem that made him famous: if a hexagon is He derived it simply and directly from Pascal's (1639) theorem: if
a hexagon is inscribed, then the points common to opposite sides are collinear. It was soon discovered that whenever, in a theorem, each figure was replaced by its reciprocal, the resulting statement not only made sense but also proved to be true.
Another famous example is due to Gergonne (1825).
He found that
Desargue's (1648) two triangle theorem yields its own converse.
Gergonne
suggested that what is involved here is a general principle of geometry that does not depend on the mediation of a conic, and he named it the principle of duality. A controversy arose, and it was particularly bitter between Poncelet and Gergonne.
[ES
Poncelet rightfully claimed priority for having developed a general
theory of reciprocal figures (an 1824 memoir was not published until 1829 because Cauchy delayed his referee's report).
lS based on these
that a mediating conic (or quadric) is needed to justify the theory.
lllispe [1970-8J.)
The
:he lines in a plane is
The controversy was finally settled with the development of analytical projective geometry.
i on the following
calculus.
Le is inscribed in a
First, Moebius (1827) introduced his barycentric
Then, independently, Plucker (1829, 1830) introduced the idea, which
is in common use today, of homogeneous coordinates
£
common point of two
representing points, and
diagonal points always
symmetric relation of incidence,
lint
P
P; if
P
gives rise to
.lled the pole of on
M
M is equal to
In sum,
that preserves
: 2.
=
C
(qo' q1' q2)
representing lines and, the
o Plucker's explanation of duality is essentially this:
Viewed as the algebra of
dual triples, the geometry of points and the geometry of lines (considered as
:ate about a certain
R.
(PO' PI' P2)
is
,ed to vary along a
P
=
is obviously the
through
Gergonne,
on the other hand, could not justify the principal of duality mathematically.
,veloped for over 2000
L
Poncelet, moreover, insisted
gives
elements) are indistinguishable.
Moreover (and much more will be said about
this below), a given curve may equally well be described by an equation relating the coordinates of its points or by an equation relating the coordinates of its tangent lines. The duality of curves cannot be considered as perfect unless the following reciprocity theorem holds: if
X'
is the reciprocal of
X , then
X
is the
STEVEN L. KLEIMAN
166 reciprocal of
X' .
In the case of polar reciprocation,
X'
particular,
is, by
X; an envelope is the
definition, the envelope of the polars of the points of
locus of the "ultimate" intersections of a series of curves, so a point
X·
is the limit of the point
approach each other. p'
X , whose end points approach each other.
X.
curve
X
is the pole of a tangent of
X'
construction of the envelope
X"
of
p'
P'e
X'
of
X'
is the envelope of the tangents of
of
X'
p.
is, by definition, the envelope of X"
Xn
One standard counte
tangents pass through a
X has equation
that
parallel to the x-axis;
X
Now
X.
X"
Likewise,
is the envelope of the tangents of
X.
f
Hence, once
Monge did so, interestingly, in connection with a
method, going back to Legrange and Legendre, for solving certain kinds of (slope, negative-y-intercept)-locus (m( x), b(x»
X: f(x,y)
To a curve
0,
=
X· ; that is,
Monge associated its
is the locus of the
X'
where
(I)
THEOREM (Samuel
(3)
either
and
Monge proved the reciprocity theorem, that
(2)
b
X"
IDX
-
Y
X, as follows: (J) yields
is a line,
0]
The smooth conic il following reason as weI' that every tangent pass, infinitely many plane obviously, if
X'
CI
is a
P
corresponding
in §I-3.
It is unclear
seem that there are no
1-2.
hence,
BASIC CONSIDERATI
algebraically closed of db/dm
(db/dx)/(dm/dx)
Thus the reciprocal of
X'
Monge's definition of provided only that
x
is equal to X·
and
x
(db/dm)m - b
y
X
makes sense in arbitrary characteristic
separable over that of
is a separating transcendental of the function field of
and
db/dx
X' .
X is
Indeed, the separability means precisely that
cannot both be
0, so (2) implies that neither is
otherwise
fied.
M
subvariety of
JPN, am
p
His proof of reciprocity works, provided that the function field of
dm/dx
such that
reciprocity fails.
( dm/dx)x
db/dx
JPN X
point dy/dx
m
X"
The above example
Therefore, in both cases,
Another process of reciprocation had already been introduced by Monge in
differential equations.
Indeec
X
curve in
reciprocation for curves.
X.
f
the center of a conic a!
1805, the year before Brianchon and Livet initiated the theory of polar
X.
0
reciprocity fails;
of its tangents.
point
A better name
X.
"center"
is the locus of
X is equal to the envelope
the reciprocity theorem says, in other words, that
infinity on the x-axis. of
X, and since incidence is symmetric, the points
points of the original plane representing the tangents of again,
theorem in a similar way
The rec iprocity the
Therefore, by what was just said,
correspond to the tangents of
X with!
reciprocal of
defined the reciprocal c
In other words, the
is, by definition, the envelope of the lines of the dual plane that
correspond to the points of
m is also
is not hard to
is just the polar recip!
Hence, by
In the case of abstract duality, the situation is exactly similar.
X'
is
is reciprocal to the rectification of the
Now, the reciprocal
the polars of the points
of
p'
of intersection of two polars as the two
Since polar reciprocation preserves incidence,
the pole of a cord of continuity,
PIe
It
0; in
The smooth locus of X A rigorous general best developed using th CX
of
x: is defined a
AND
x'
particular,
is, by pI
of
167
m is also a separating transcendental, so
X"
is defined.
It is not hard to see, that Monge's
X ; an envelope is the eves, so a point
DUALITY
reciprocal of
reciprocal X' is equal to the polar X with respect to the conic 2y = x 2 , if p # 2 . Monge also
:wo polars as the two
defined the reciprocal of a surface in 3-space and proved the reciprocity
!s incidence,
theorem in a similar way.
other.
pIe
is
Hence, by
Chasles (1832) pointed out that this reciprocal too
is just the polar reciprocal with respect to an appropriate quadric. The reciprocity theorem is not always valid in positive characterisitic
:her words, the
X when
tangents pass through a particular point P. Indeed, choose coordinates so that X has equation y K x 2 . Then dy/dx = O. So all tangents are
Xli
p
All its
.nition, the envelope of lat was just said,
One standard counterexample is the smooth conic
= 2.
:ectification of the
p.
parallel to the x-axis; in other words, they pass through the point ,xactly similar.
Now
infinity on the x-axis.
X.
dual plane that symmetric, the points
"center" of
f
p
is sometimes called the "strange point"
A better name was suggested by Graham Higman (pvt. comm.): the
of
II
The point
X.
Indeed, it is traditional in projective geometry to define
is the locus of
the center of a conic as the pole of the line at infinity.
X'
reciprocity fails;
Hence, once
certain kinds of onge associated its ls the locus of the
The smooth conic in characteristic whenever
p
2
obviously, if
P
X'
is a line, then every
corresponding to
is a special case for the
X'
In fact, there exist is a given plane curve;
X passes through the
tangent of
X' , and conversely.
Such
X will be constructed
X' , when there exists a smooth
It is unclear, given an
Then,
p > 0 , there are singular curves such
X such that
infinitely many plane curves
p •
X is a (plane) conic.
is 2 and
that every tangent passes through a given point.
in §I-3.
follows: (1) yie Ids
X be a smooth, irreducible, closed
Let
such that every tangent passes through some point
following reason as well:
point y
]I' N
X is a line, or the characteristic
either
theory of polar in connection with a
At any rate,
P !
THEOREM (Samuel [1960], p. 76) .
(3)
curve in ntroduced by Monge in
X"
The above example is rather special; witness the following lovely result.
refore, in both cases, s equal to the envelope
at
P
X.
It
seem that there are no other examples known of a smooth plane curve for which reciprocity fails.
1-2.
BASIC CONSIDERATIONS.
From now on, the ground field
algebraically closed of characteristic m - b
y
otherwise specified. subvariety of
characteristic Dction field of
X is
The smooth locus of
0; in
will be
p, which will be arbitrary unless
X will be a reduced and irreducible, closed
and n
:=
dim(X)
X will be denoted by
> Xsm'
A rigorous general treatment of the duality of projective varieties is
ans precisely that neither is
,
p
the function field of
t
]I'N
Moreover,
k
best developed using the (projective) conormal variety.
ex
of
X
is defined as follows.
First, let
]I'N*
The conormal variety
denote the dual projective
STEVEN L. KLEU1AN
168
space, whose points represent the hyperplanes of WN ; if WN* = W(V*) where V* denotes the dual vector space.
WN
= W(V)
,then
no
[1956J found that, alth'
p'
WN*.
of
The notation
pI
characteristic, neverth,
P
of
WN
and
Ht
is
A similar convention will be used for a point
and the corresponding hyperplane
consistent with the notation X' introduced below. Define ex sm as the set of point-hyperplane pairs (p, H) such that P lies in Xsm and H contains the (embedded) tangent space TpX It is clear that ex sm is of dimension N-1 ; whence, so is ex. Let I denote the graph of the point-hyperplane incidence correspondence; that is, of point-hyperplane pairs
ex
is contained in
*
lies in
P
H.
I
is the set
X'
of
X
is defined as the image of
It is, obviously, a closed subvariety.
reflexivity.
Reciprocity
is defined as the condition that
other words, reflexivity means that a hyperplane
P
iff
Let point
p'
H
Q must lie on
Then
However,
previous articles, contribution; certainly
(4)
THEOREH (The Honge"
ex -) x'
is genericall:
If the characteristic
The second asserti,
X'
X"
=X
Q'
generically smooth iff
is the
The proof of the f
ex'
and remember when it is geometry.
In
Briefly put,
Consider dual systems
0
X at a
at the point
is tangent to
X'
(qo
1.
H' at a
Then the graph
I
of t
H, because incidence is preserved.
Q need not be a point of contact of
correspond bijectively and
ex
is tangent to
is a hyperplane that is tangent to
Q be a point such that the hyperplane
H'
criterion was, regretab"
Then, obviously,
condition that X" = X (a meaningful condition, once WN** is identified with WN ). A closely related but more useful condition is that of
point
following useful criter:
standard basic theorem' or reciprocal,
WN
in
such that
I
The
ex
CP, H)
he did n,
is
likely to result, the same letter H will be used to denote a hyperplane of WN and the corresponding point of WN* ; otherwise, H' will denote the corresponding point.
dimension over the comp:
H , even when
X and
X' Differentiating yields
; witness the next example.
(5)
Reflexivity obviously implies reciprocity. p ) O.
The converse may fail when Now,
A standard counterexample (Wallace [1956], last section) is the smooth of
curve, x q + 1 _ yq
X
y
= xq
and
where
=-y
yq
x' is defined by the same equat ion as
X.
Indeed, m
dy/dx
not reflexive.
Indeed, if
corresponds to the point H'(xq,yq) unequal to
b
H
So
corresponds to the point
of
X·
b
of
ex -)
However,
X
p(x,y) , then
H
X·
is
Note, moreover, that
X -- and note that here the
X' , is isomorphic to the Frobenius endomorphism of
X .
Monge's analytic treatment of reciprocity was generalized to arbitrary
of c W
simple, but useful, cr: variety of its image projectivized conormal
e
at
X , is an isomorphism -- in fact, it is obviously an
isomorphism for any smooth hypersurface curve
ex -)
that is,
X , which is usually H.
Y
the present case,
X' , and the tangent to
Q(x q ' ,yq2)
is canoni
smooth variety mq + 1-b q + 1
x
X"
X at
P(x,y) , although it is another point of
the projection, projection,
pe)
Hence,
is the tangent to
H'(xq,yq)
q
I
(resp. c f
pN
is an (N-1)-dimensi,
precisely, wlC
=0
)
In the present case, criterion, so with res
ex 1-3.
ex'
provided
ex
NONREFLEXIVE CUR
field extens
k(CX
AND
if
JPN '" JP(V)
, then
wnen no confusion is
-tat ion
p'
dimension ove, the complex numbers by C. Segre [1910], §4, although (probably inadvertent
he did not mention Monge's name.
Nearly 50 years later, Wallace
characteristic, nevertheless, the proof in characteristic 0 leads to the
will denote the
for a point
169
[1956J found that, although reciprocity is not always valid in positive
enote a hyperplane of i'
DCALITY
P
of
JPN
and
H'
is
following useful criterion of ,eflexivity in arbitrary characteristic.
(The
criterion was, regretably, not named after Monge in the present author's previous articles, because earlier he was unaware of the extent of Monge's
(p, H)
such that
is clear
It
Let
denote the
I
that is, H.
P
Reciprocity
is the
is identified
n is that of ,at
ex
=
ex'
tangent to
X at a
tangent to
p = 0 , then
is reflexive iff
X is reflexive.
The second assertion of (4) follows from the first.
generically smooth iff the function field extension is separable. The proof of the first assertion of (4) is easy to understand, appreciate and remember when it is developed from the point of view of Lagrangian Briefly put, the idea is this (for the details, see Kleiman [1984]). Consider dual systems of homogeneous coordinates for JPN and JPN* , and
at a
Then the graph
I
of the point-hyperplane incidence correspondence is given by
is preserved.
when
X and
o
I
X'
ample.
Differentiating yields the relation,
Now,
section} is the smooth of
JPN
I
is canonically isomorphic to the projectivized cotangent bundle
(resp. of
smooth variety
)
However,
X
P(x,yl , then
H
agent to
Xf
is
at
Note, moreover, that
The projectivized cotangent bundle
w = pdq
variety of its image
(resp.
V ln
CX
of
PT*
criterion, so with respect to
1-3.
provided
CX -) X'
qdp
by (5).
In
is the conormal
C
is a Lagrangian (that is, w
o (more
C -> V , is generically smooth. by the
Hence, by the criterion again,
is generically smooth.
KOKREFLEXIVE CURVES. Assume that
field extension,
w
is the closure of the
is a Lagrangian with respect to pdq
GX = CX'
,lized to arbitrary
C
is an (N-I)-dimensional solution of the differential equation
In the present case,
of any
:n the general case, there is a
Vsm ) iff (a)
precisely, wlC = 0 ) and (b) the ?rojection,
note that here the X .
w = qdp ).
Y (that is,
projectivized conormal bundce
it is obviously an
1domorphism of
PT*
N carries a canonical contact form
simple, but useful, criterion: a closed subvariety
C
which is usually
pN*).
Y of dimension
the present case, that is,
o
pdq ... qdp
(5)
verse may fail when
x
Indeed, it is a
standard basic theorem of the theory of smoothness that a map of varieties is
H'
X'
X
smooth (that is, smooth on a dense open subset).
geometry.
In
at the point
"
is generica)
If the characteristic
Then, obviously,
'ined as the image of pN**
THEOREM (The Honge-Segre-Wallace criterion)
(4)
ex -) X'
is the set
I
contribution; certainly, it is no less important than Segre's or Wallace's.)
X
is
k(GX)!k(X) is trivial, and if
plane curve. X
Then the funct
is reflexive, then
STEVEN L. KLEIMAN
170 k(CX)/k(X')
is also trivial.
in §I-I), if k(X)/k(X') is separable, then extension is birational.
given separable and inse
Hence, by (4) (or by Monge's own version of it
The lattice of fiel
X is reflexive; whence, the
On the other hand, any given power
possible inseparable degree, and any given
>
s
q = pe
that
is a possible separable
I
q > I ; that is, that there is some inseparability.
degree, provided only that
In fact, it is possible (compare Wallace [1956J, last section) to construct an appropriate
X'
X such that
is equal to any given plane curve
solving Monge's differential equation (I) as follows. m, b
transcendental for
k(Z) = k(m,b)
extension of
of degree
k(m)
k(rn,b) / kern)
Z so that
for
over
k.
(Such extensions exist.
generated function field
of degree
containing the
algebraic closure of the prime field and given any finite separable extension L
of
M ,
there exists a separable extension
K of
may assume first that extension of
k
over
=p
), noting that the compositum of all Kummer extensions
(resp. of all Artin-Schreier extensions) is of infinite degree
K '" k(m)[ z]
Replacing
m is separable over k(m,z,b)
k(x)/k(z)
by
z + m if
(Indeed, if
rn, then
is a separable extension of k(z).
k(m,b)
m
over
k( z) •
kern,z,b)
xq
Say
=
z.
hence, so is
q
and
bare sep It
follows
(6)
Indeed, raising the equa Hence, over
k(yq,z) k(z)
x
is cont
but insepan
view of (6),
plainly
is one for
yq .
k(x,y)
Hence, there are
F
D(x q
0 .)
Then
Say
x
F(x,y)
o.
k(m,x,b)/k(m,b) of degree
kem,x,b)
However,
q+ 1 • F
cornm.), if
k(x,y) k(m)
Thus, bot
is a polynomial
0
-F 2 (m,z)[dm/dz)
FI(m,z)
is purely inseparable of degree
=
k(m,
Interestingly, bee
dz/dm '" 0 , we may assume
F(m,z)
over
q.
Now,
K will do the trick.) z
k(z)
relation of minimal degree in Then
s
M is a pure transcentental K, use Kummer theory if s F p (resp. Artin-
M; hence, one such extension Say
that
s
of any given degree
Indeed, observe that we
is a prime and then that
To construct
Schreier theory if s of degree
s
M
LIM
that is linearly disjoint from the extension
q
the other hand, its degr which is
In fact, given any separably k
k(m,b)/k(
one hand, it is purely i
K be any separable
M in one variable over a field
course, defined by (I); z = x q is separable ove
Indeed, assume (i),
m is a separating Let
is a separating
unlikely that
(The present version of
that is linearly disjoint from the extension
s
x
be a separating transcen
Z by
the construction was worked out in collaboration with Thorup.) Choose coordinates
X,
for any plane curve
is a
(Here
rnay have
P '" 2 , thE
k(m,z) X
Hence, k(m,x,b)/k(m,b) Since and dy/dx Thus
b
m and
b
is of separable degree are separable over
are separable over lies in
k(x,y)
=
k(x,y) . kem,x,b) .
of a plane curve
k(x) Hence,
and
X' '" Z
s
k(z)
x
and
and inseparable degree but
dm/dx '" 0
m lies in
Therefore,
X such that
is such that
k(x)
k(z)
and
x
is inseparable, dm/dx '" O.
k(x,y) ; whence,
b
However, does too.
yare the coordinate functions
and such that
k(X)!k(X')
The last example
q m
is of the
P '" (y3,
(7)
THEOREM (Pardini
curve of degree p+ 1 , then
d
in
X is proj
The key to Pardin
TANGENCY
'nge's own version of it q
= pe
is a
for any plane curve that
is some inseparability.
;ection) to construct plane curve
by
Z
171
sand
q.
The lattice of fields above is not special to the construction but obtains
a possible separable
1
DUALITY
given separable and inseparable degrees,
exive; whence, the power
AND
x
X , provided: (i) the coordinates
is a separating transcendental for
course, defined by (I); and (iii) z = x q is separable over k(m,b)
separating
unlikely that
k(m,b)/k(m)
and
pe
is taken with
over
q
k(m,z,b) .
e
k(m,z)/k(m)
Consider the extension
the other hand, its degree is, obviously, at least that of
k
which is
containing the
:e separable extension of any given degree
s
q.
a pure transcentental
f
8
-# p
(resp. Artin-
f all Kummer extensions 8
of infinite degree
m and
b
over
k(z).
It follows that
are separable over
(6)
Hence, over
is a polynomial
,z)[dm/dz] -# 0 .) q c = z. Then
) is
k(m,x,b)/k(m,b)
k(yq,z) k(z)
y
k(x) .
= rnx-b
is contained in
plainly
is one for
z
yq.
X
O.
q+ I. F
comm.) , if
However,
m and
b
Then
F
'nce,
b
However, does too.
q.
are separable
(Here
x
and
=
b
=
mqz-b q .
are separable Finally, in
k(m,z,b)
because
P =
curve of degree X
xq ,
D, M, B
such that
yare, of course, viewed as indeterminates.) P.
For example (Hefez, pvt.
2 , the smooth conic
o
(y3+ I )y_(x 2 + l)x =
THEOREM (Pardini [1983])
p+1 , then
m and
yq
Dy - Mx + B
The last example is special to (7)
b
divides the polynomial
may have smaller degree than p
is such that
'dx = O.
One
are rational funtions of
X
ill
=
k(x,y)
F(x,y)
However,
is inseparable,
and
Therefore, (6) holds.
Hence, there are relatively prime polynomials
of degree
q
m
is a separating transcendental for
P
,eparable degree
z
On the
k(m,x,b)/k(m,z,b) ,
to the qth power yields
k(yq,z)
Interestingly, because of (6),
Say
Hence,
k(m,z,b)
but inseparable over
view of (6),
dm = 0 , we may assume
=0
because
q
k(m,z,b)
Indeed, raising the equation
x
k. )
is inseparable
Thus, both extensions are purely inseparable of degree
Now,
ldeed, observe tha t we
.8
m need
k(m,x)/k(m,z)
oint from the extension
Id
minimal such that
k(X) = k(m,x,b)
one hand, it is purely inseparable of degree at most
any separably
are, of
are necessarily linear disjoint.
Then
e any separable
n
b
k(m,z,b) , and even if it is, it seems
Indeed, assume (i), (ii) and (iii). of degree
m and
However, it seems unlikely that
be a separating transcendental for
(The present version of 'horup. )
=
q
x, yare chosen such
k(X)/k; (ii)
d
Let
(x+y)F p
=
2.
Indeed, witness this:
X be a smooth but not reflexive plane
in characterisitic
p> 2.
Then
is projectively equivalent to the curve
plCd-l) , and if y
=
d
xp+l_yP .
coordinate functions
:)/k(X')
is of the
The key to Pardini's proof that
p:(d-I)
was this discovery: if
X is
STEVEN L. KLEIMAN
172
defined by the homogeneous polynomia: .
derivatives
F , then all the various second partial
are identically O.
,J
(Hefez ([1984] (7.5, ii)
remarked that
F
is not reflexive, as theorem, ql(d-1)
(d-1)F i = 0
pl(d-j)
Then the
If
= 2z P
# 0.)
L
Now, by Euler's P
Pardini's second assertion is deeper, and its proof is nonconstructive.
X into
p8
via the coefficients of
X
H
X
Then each orbit is 8-
is finite.
dimensional and constructible; hence, it contains a dense open subset of the Therefore, any two orbits must intersect, and so coincide.
is a linear space
= yPx-x P
into
y
xp+J_yp.
If
X, although
plane, and as before, let inseparable degree of projection, tangent
CX
H of
sand
To get a
k(X) = k(CX)
over
Pardini's work
x' , consists of
s
is tangent to
at
X
X
s
P.
X
The guess is borne out by the earlier example, R
H : y
and it is easy to check by direct computation that the
xoqx-yoq
intersection cycle
X at
[R.X]
P(xO'YO)
is given by the equation
because
q
H
I , then
is tangent to
iCp, R.X)
X at q.
P
Thus
Thus, set-theoret
in other words,
Xl is
discriminant is not 0: i(P, H.X) = q
direct computation using the relation (6). conceptual proof will be given below.
P
hypersurface, then thl
xo'Yo)] + [Q(xOq2'YOq2)]
In fact, it is not hard to verify that
X at
hyperplane sections oj
is given by
[H.X]
P.
be a poin
(8) indeed,
the tangent line =
of
P
tangent to
(P, R.X) ,
J -yq
y
X
exactly along a linear Let
q.
is equal to the inseparable degree
is not ref
X
is never reduced.
It is now
natural to guess that the order of contact, the intersection number
v
XH
precise, just when
Rence, a general
distinct points
t
Jacobian ideal of X (l
X need no longer be
distinct points.
then
If
Then a general fiber of the
k(X')
FOI
2d-fold Veronese map, out 2V
denote the separable degree and
q
X is reflexh
be about anything.
was generalized to higher dimension by Hefez [1984], §§7-9; see (14) below. Consider still the case of a curve
"Ii
smooth hypersurface of
feeling for the content of the assertion, try to find an explicit linear y
[J907], Ch. 9,
discuss reciprocity not
D, M, B above, and to show that the linear group of
transformation that carries
is reflexiv
linear space; in fact,
Bertini
acts so that the stabilizer of any
X
introduced
above turned out often to be
the linear polynomials
(CXl H ,
It would be good to know whether
It would be nice if the polynomial
The idea is to embed the space of all
H-contac
may be viewed as a clos
0
however,
whence,
q > P
also when
0
dy/dx
zp-2 y 2 - x p
-
H be a hyperplane
H.
theoretic fiber
smoothness is necessary here; for example, the curve
x
Let
discriminant would be
whenever
q
I
locuS.
by a
However, a simple and ffiore general,
Now, if
q >
iCp,
H.X)
>
q
iff
X
of this section. Equation (8) hoI
2 ,
X is reflexive, that is, if
Hence, if i(P, H.X)
or not,
is not reflexive.
Some examples
the scheme structure differentials.
Indee
There is a higher dimensional version of this theorem, (10) below; before it
equation:
can be discussed, the notion of contact locus must be introduced.
(9)
1-4.
To prove (9), pick dl
THE CONTACT LOCUS.
Let
X be of arbitrary dimension n'
dimC X' )
n , and set
and
as in
§
1-
TANGENCY Ie various second partial (7.5, ii»
remarked that
Let
H be a
H .
Then the
Now, by Euler's
e good to know whether
P
olynomial
If
Since
CX
is ref:exive, and if
X
.a the coefficients of Then each orbit is 8-
If
X
is reflexive but
be about anything.
2d-fold Veronese map, then smooth hypersurface of
In explicit linear
out 2V
l_yp.
Jacobian ideal of
Pardini I S work
)7-9; see (14) below.
X is reflexive by (20) below.
If
2V
m a general fiber of the
exactly along a linear space!
Hence, a genera 1
is never reduced.
Let
P
tangent to P.
and if
d
H
If
V
is any
is the hyperplane that cuts 2 ;
indeed, the
is obviously generated by the equation of
V
is not reflexive, then, by the Monge-Segre-Wallace criterion (4),
X
precise, just when
indeed,
but does not
V as schemes by (8) below, at least if p
XQ H
XH
: y
X'
X' , then XH may is a projective space embedded by the
X
X of degree
.ble degree and
,ction number i(P, H.X) ,
Bertini considers
H' is not a simple point of
X need no longer be
is now
°
For example, if
,coincide.
To get a
p =
discuss reciprocity nor reflexivity.
lse open subset of the
It
is a
X' , then
[1907], Ch. 9, n. 13, p. 200, gave an interesting geometric proof that
is a linear space when
at the linear group of
P.
is a simple point of
H'
XH
X.
introduced Bertini
)ints
representing
linear space; in fact, clearly,
of is nonconstructive.
"
WN*
XH ' is defined as the schemeis contained in XXX' ,obviously
may be viewed as a closed subscheme of 2 # 0.)
173
denoted
H-contact :ocus (CX)H' .
DUALITY
H' denote the point of
and let
theoretic fiber
AND
X
It is indeed a striking
be a point of
X at
P
fact that as a rule--to be
is reflexive--a general tangent hyperplane is tangent
Xsm
Then, obviously,
(that is,
H contains
XH iff H is X 0 H is singular at
P lies in
TpH), iff
Thus, set-theoretically,
(8)
Singe
he equation
in other words,
putation that the
hyperplane sections of
xsm
Q H )
X' is the closure of the discriminant locus of the family of Thus, if
Xsm.
hypersurface, then the projection
CX ->
X is smooth and
*
WN
X'
is not a
is a proper map whose
discriminant is not of codimension 1; if the
were finite, then the
discriminant would be of codimension 1 by the theorem of purity of the branch
rhenever
q
2:.
by a
simple and more general,
or
not,
locus.
Some examples will be discussed below after (12) and again at the end
of this section.
i(P,H.X) > 2,
Equation (8) holds scheme-theoret
if the right hand side is given
re flexi ve, that is, if
the scheme structure defined by the (n-1)th Fitting ideal of the sheaf of
X is not reflexive.
differentials.
: 10) below; before it
ltroduced.
lion
n, and set
Indeed, (8) is the result of taking fibers in the following
equation:
(9) To prove (9), pick dual systems of homogeneous coordinates and
N W
*
as in § I-i.
Set
ill
=
N-n , and let
Fl' ... , Fm
and
L
for
N W
be homogeneous
STEVEN L. KLEIMAN
174 polynomials that define
X
in a neighborhood
U of
P
DO' ... , DN and consider the
Let
denote the partial derivative operators with respect to
c
I.
Then
V
is a cc i (
maximal minors of the matrix, iff
[
The hypothesis
DOF;
the same dimension, narnel:
PN]
component of DOFm
DOFm
eXsm Q (UxF N*)
UxF N*
in
,
other hand, the minors define the singular locus of the
(N-I)
if
c : c(X)
m+I
equations defining
>
n
space
T
to
contained in
c = 0
iff
One is
over
>
hyperplane tangent to
X at
P
is a curve but not a line, then
P
of
X'
Here is a remarkable fact: if
Plainly,
I-S.
X not in T
T
X
T , then there is a So, in particular, if
X
the next theorem, (IO), the only smooth
0 < c < n ,
X
is a curve.
Hence, if
X
In particular, in
THEOREM (Hefez-Kleiman [I984J, (3.5»
hyperplane, and
Let
THE HESSIAN.
ex.
SE
This invariant might simple point, then defined at P
in
h(P,H)
X by th
regular system of parametE differentials
dti
form c
corresponding first partie ( I ])
h(t
Indeed, in view of of
XH .
X il H
in
X, it is c
X by the vanishing of
H be a general tangent
V an irreducible component of the contact locus
The
Denoted
h(P,H)
X of interest is a curve.
The general result on generic order of contact (see 1-2) is this: (IO)
=
c
ex.)
Indeed, consider the tangent
is smooth and reflexive and
c = I , then
(In Hefez-Klei11
function on
is even; see (24) below and the surrounding discussion.
is smooth and reflexive and if
Therefore, the
out a curious fact:if
is a hypersurface. X
P.
computational, longer and
o
not containing
is smooth at
Q p'
reduced along
They are equal, on the other hand, iff
c = O.
X'
complete.
X , because
fails to be a hypersurface.
c
(XxX')
(P,H) be a general pc
passes through the point So
is a generic tangent hyperplane.
X is a linear space.
because, if there is a simple point
c
ex.
X at some simple point. Clearly, the dual linear space T' is X' The two are equal, on the one hand, iff they have the same
dimension; that is, iff
then
X'
the intersection, let
(N-I) - n'
is a measure of the amount by which
Xsm. because both set
scheme equation will resul
might be called the duality code feet of n - c
Notice that
H
The equation in quest on
question is equal to the 11
cod C ex, xxx' )
cod ( XH ' X )
appei
H
known (and follows from
FN*-scheme,
There are two important numerical invariants associated with
This invariant
On the
0 U] x F N*) 0 I
because the matrix is just the Jacobian of a system of the scheme in UxFN*
n'
Q
=
the right side of the equ.
because
the remaining rows, which is simply the conormal space in question.
C[x sm
X
c
equation in question is a
their vanishing is just the condition that the first row belong to the span of
n +
is not reflexive.
Po
On the one hand, these minors define
c
X
v.
Assume
f
sheaf--cotangent sheaf sec
AND Let DO' '" , DN , and consider the
c = I .
Then
V
is a component of
X
DUALITY
I)
175
H , and [ k( CX)
iff
X
k( Xl ] insep
is not reflexive.
The hypothesis
c
I
the same dimension, namely, n-I . component of
X
I)
XH and X I) H be of dimeXH) < n-l , then each
is just the condition that Note that, if
H appears with multiplicity
The left side of the
equation in question is always at least 2; whereas, if .n
UxlP
N*
,
the right side of the equation is trivially equal to 1 •
because
" belong to the span of in question.
On the
The equation in question is equivalent to the scheme equation X I) H XH Xsm , because both schemes are divisors on Xsm and because, as is well
on
known (and follows from
EGA IV 3-9.8.6, p. 86), the inseparable degree in question is equal to the multiplicity of appearance of V In XH . Now, the scheme equation will result on taking fibers if ex is a scheme component of
JPN* -scheme,
m+1
equations defining
the intersection, let
lated with ['
ex.
One is
(XxX')
I
I)
passes through the point X'
over
I)
is smooth at
p'
P.
H'
CX
of
H'
Then the hyperplane
X'
but
However,
complete.
ldefect of
computational, longer and more involved. reduced along 1-5.
ex.
eed, consider the tangent
I , then
c
T
in particular, if
Denoted by h(P,H)
:uss ion.
X
simple point, then X
0 < c < n , Hence, if
defined at P
hX(P,H)
X
In particular, in
hep,H) , it is defined by
n
dim(QICX/x,(P,H»)
n
dim(Q1 X/k (P»
in
h(P,H)
may be calculated as follows.
X by the vanishing of the function tl' ... , tn
for
X
at
f
Say on
P
X X
( 11)
P I)
is a
H
is
Choose a
(that is, the
dt i form a basis of Q1X/k at p). Denote by corresponding first partial-derivative operators. Then,
of
general tangent XH •
is
the
h(P,Hl Indeed, in view of (8) and of the conormal sheaf--cotangent sheaf sequence
1-2) is this:
let locus
or simply
differentials
ls a curve.
t
(XxX') 0 I
This invariant might be called the Hessian rank, because, if
regular system of parameters
lexive and
Moreover, the present proof brings
H
, then there is a
0,
(XxX') 0 I sm CX , and the proof is
X is not reflexive iff
iff they have the same other hand, iff
is the fiber of
The second invariant is a lower semi-continuous numerical
is
T'
H' .
ex .)
THE HESSIAN.
function on
1 1 inear space
at
X'
(In Hefez-Kleiman [1984J, there is another proof, which is
out a curious fact:if Plainly,
X' 0 p'
*
lPN
of
p'
p' is not tangent to
Therefore, the latter is reduced along
ic tangent hyperplane. X , because
X is not reflexive,
To prove that it is, if
(P,H) be a general point of
So
)
lypersur face.
X is reflexive, then
X0 H
in
X , it is clear that
X by the vanishing of Assume
f
and of the
sheaf--cotangent sheaf sequence of
XH is given in a neighborhood of P in Dif In view, now, of the conormal
XH
in
X, it is clear that
h(P,H)
is
STEVEN L. KLEIMAN
176
equal to the rank of the Jacobian matrix of h(P
f, D1f, •••
and that (P,H)
=
h(P,H)
0
f
x
Moreover, all
X
of de!
The principal new
d'y/dx 2
(see the
TOFO(TOP,
(1936), which asserts tl
1-2), the condition is, in fact, necessary and
exists another one
Indeed, the next result gives the general form of
obtained from
this criterion, a criterion for the reflexivity of a hypersurface. THEOREM (Wallace [1956], 6.2).
If
X
:8
C
(14)
is contained in the union of the
is a separating transcendental for
(13)
To prove (13)(i) =>
case (i) gives a
Monge's original proof of reciprocity for a plane curve (see
I O.
nonsingular.
11-3.
in characteristic
1
Pardini's theorem (
In each case, there is an
The elementary method used by Landman
c < 2m ; indeed, obviously, n'
c.
c . 2m
X,
(There is another way to get these lower bounds; use 11-(22).)
more general bound.
form a regular system of
Then
X (13)(ii), take the open set to be the intersection of Xsm
and the open set of (12) (in.
the
Consider
Fix
would be O.
P
and reorder
t I' ••• , tN
H
aN" 0 , because, otherwise, ai
would be 0, whence all
aldt] +
Now, (11) applied to
f '" altl +
h(P,H) Hence, (13)(ii) holds because (12)(ii) does.
x
(see the
Of
course, the implication, (12)(iii) "'> (12)(ii), is trivial. Pardini's theorem (7) concerns
> 2 that is not reflexive. (14)
Since
a smooth plane curve in characteristic n '" 1 , nonreflexivity is the same as the
being identically O.
h(P,H)
Hefez generalized (7) as follows.
THEOREH (Hefez [1984], (7.7), (9.11».
hypersurface in characteristic
p > O.
Assume that
Then the function
identically 0 iff there exist homogeneous polynomials
is a normal
X
h(P,H)
FO' ... ,FN
is of the
same degree such that
x Horeover, all
d 2 y/dx 2
fact, necessary and
Similarly, the implication,
(13)(iii) => (13)( ), follows from the implication, (12)(iii) => (12)(i).
shows
§ I-I)
Dcity obtains if
n
at
nonsingular.
function
case, there is an es
X and
of X
f 1 ines in an
0
:ker map.
P
for
is nonsingular.
form a regular system of parameters and such that, if
ls-Harris [1979]): (i)
!smannian
I' ••• , t N
corresponding partial-derivative operators, then
Then
is not a
X'
coordinate functions
appropriately.
consisting of points
[(DjDitN)(P)]
(iii) There exist a simple point
lower semi-continuity, the ugh possibly it is empty.
X
such that, given any system of inhomogeneous coordinate functions
partial-derivative operators, then
that
177
is reflexive. (ii) There exists a dense open subset of eX sm consisting of points
is veri fied.
ry
DUALITY
: (i)
following statements are
) (p) ] I
AND
X of degree
p+l
given as above are projectively equivalent.
The principal new ingredient in the proof of (14) is a theorem of Hasse (1936), which asserts that,
a nonsingular (N+l)x(H+I)-matrix
exists another one
T '" T(P)B,
T
such that
where
Tep)
is the matrix
ives the general form of
obtained from
fpersur face.
proof of the second assertion; that proof is fairly involved.
T by raising each entry to the pth power.
It is used in the
Here is the idea of the proof of the first assertion of (14).
:surface, then the
x
F
o , and
fix
(P,R)
in
B, there
Say
Note that the proof of (11) may be
p
STEVEN L. KLEIMAN
178
the following version of
applied to h(P,H)
=
XH viewed as a subvariety of H ; it shows that, because 0 , the quadratic term of F in the Taylor expansion of F at
vanishes when restricted to
H
bi .
( 14' )
Fix on
X
P i
, and let
( 14 T) yields deg(Fii)deg(F)
X
X
is normal, there exists a ordV(F i ) .
If
> Iv ordv(Fii)deg(V) Fi
deg(F)-I. or
0
Fj
zero polynomial, for all ORDINARINESS.
j
,
i
Fii
>
=
or
0
and
j
.
Fij
X'
=
Fii
X'
It is evident that
= 0
on
X.
Hence,
Hence, by Fij
The first assertion of (14) follows. So, following Hefez-Kleiman
X
->
x'
CX -> X'
is a hypersurface iff
PROPOSITION.
On the other hand, if
is obviously birational.
XH ) X
is
at
P
by the vanishing' (iii)
that
The following corol 2, then
X is not refle'
COROLLARY (Katz [I
(18)
and if
X is ordinary,
Indeed, by (In, th whose diagonal terms van
X is ordinary iff
CX -> X'
CX -> X' is birational, iff
Although, i f
X
is ordinary iff
k(CX)
k(X') , iff
k(X)
hep,H) is a finite
k(X') •
Here is the result again, expressed in more down-to-earth geometric terms: PROPOSITION.
2
p
nevertheless, in practic Hefez-Kleiman [1984], (5
and separable extension of
(15b)
There exists
X 0 H has a nonde;
Here is the same result expressed in more arithmetic terms: PROPOSITION.
is
of one nondegenerate doul
Hence, the Monge-Segre-
is generically etale.
(15a)
X
(ii) A general tang-
is generically
Wallace criterion (4) yields:
(IS)
THEOREM (Katz [197.
always a hypersurface.
is a finite set, necessarily of simple points.
ex'
( 17)
that usually come up are
X
There exists
equivalent: (i)
is the
finite -- that is, iff a general fiber (namely, a general contact locus ordinary, then
So
deg(Gi)deg(G)
for all i .
0
are hypersur.faces.
[1984], (2.3), call such varieties
= O.
0 , then by Bezout's theorem,
In practice, the varieties
reflexive and their duals
h(P,H); n .
Combining (16) and
ordV(F j )
ordV(Fi)deg(V)
Ther.efore,
such that (iii)
such that
0
There exists
in the case at hand. j
X is
(ii)
F .2F .. J
+
1.
and
n
THEOREM (Hefez-Kle
equivalent: (i)
V be an irreducible component of the zero locus of
ordV(F ii )
deg(F i )
(14'), either 1-6.
F. 2 F ..
(16)
biTi )
It follows that, for all i
So (14') holds globally on
Since
However,
Fi(P)T i ) (
(
2Ff i / i F j
holds at
P
It follows on using Euler's theorem that
. . F .. (P)T·T· 1. J
for suitable scalars
a priori knowledge of
is ordinary iff, given a general tangent hyperplane H , the contact locus XH is a reduced point in Xsm , iff some isolated component of XH is a reduced, simple point.
The lower semi-continuit (19)
that, if some component of XH is a reduced point. Recall that c < n
XH is a reduced point, then every component of
and that
c = n
iff
X'
is a hypersurface.
because of the lower semi-continuity of the function
Hence,
h(P,H) , there results
PROPOSITION.
exists a point
X
Of course, some nontrivial but standard theory is required to conclude
n
X
(p ,R)
0
For example, suppos with
J
< d < e , so n
embedding.
Let
(P,R)
( 12), He fez [1984],
§
and
e
is odd, then
and
e
are even, then
cases,
h(P,H)
6, he
n , exc
TANGENCY
F
at
a priori knowledge of
P
ller I s theorem that
(16)
j
(ii)
,
X is ordinary. CX
consisting of points
(P,H)
h(P,H) = n .
(iii)
lse at hand.
The following statements are
There exists a dense open subset of
such that
There exists a point
(P,H)
of
CX
such that
h(P,H)
n.
Combining (16) and (15") and (8) and (11) yields:
, the zero locus of
F·1
ordV(F j ) = O.
So
l by Bezout I s theorem,
,g(V) all
179
n' .
THEOREM (Hefez-Kleiman [1984J, (3.2».
equivalent; (i) and
DUALITY
the following version of the Hessian criterion (12), which does not require the
that, because
)ansion of
AND
Hence, by
(17)
.on of (14) follows.
X is ordinary. (ii) A general tangent hyperplane is such that Singe xsm n H) consists of one nondegenerate double point P; that is, Xsm Q H is defined in Xsm at
P
by the vanishing of a function (iii)
that
usually come up are
'ollowing He fez-Kle iman
The following statements are
equivalent: (i)
is the
Hence,
THEOREM (Katz [1973], Prop. 3.3).
f
There exists a simple point
whose Hessian matrix is nonsingular. P
of
X and a hyperplane
X 0 H has a nondegenerate double point at The following corollary implies that, if
2, then
X
H such
P .
X is a curve ln characteristic
is not reflexive, because, as noted above, the dual of a curve is
always a hypersurface. ) X'
is gener ically
contact locus ther hand, if
XH ) X is
(18)
COROLLARY (Katz [1973J, note on p. 3).
and if
X
is ordinary, then the dimension
n
of
whose diagonal terms vanish. p = 2
Although, if
CX -) X'
and
So, n
n
is even (Bourbaki [1959], Cor. 3, p. 81).
is odd, then
X is not ordinary by (18),
Hefez-Kleiman [1984], (5.3)(i»
call
Namely, (following
X semi-ordinary if
c terms: h(P,H) k(X)
2 ,
X is even.
nevertheless, in practice, it is often the next best thing.
iff
p
Indeed, by (17), there is some nonsingular, (skew-)symmetric nxn-matrix,
ce, the Monge-Segre-
tional, iff
If the characteristic
for a general point
n -
(P,H)
in
ex
is a finite The lower semi-continuity of
-earth geometric terms:
(19)
PROPOSITION.
exists a point
h(P,H) , and (16) yield the following criterion:
X is semi-ordinary iff
(P,H)
of
CX
such that
X
is not ordinary and there
h(P,H) = n-l .
tangent hyperplane For example, suppose that
iff some isolated with
1
]pM
v
be the d-fold Veronese embedding for some If
p
2
and
or
n
is even, then
Y
]pN-I
2
p
If
and
n
is odd, then
vX
for
is in fact
Y may also
In this way, th
that
X'
xy+yz+:
w
X
is a hypersurfaci
is not reversible, and (ii P
in
]pN
Xsm.
at
Choose a system of inhomogeneous coordinates Reorder them so that
P.
a regular system of parameters for hyperplanes of
X at
P.
TI' ... , Tn
Since, under
d TI
H
through
P
such that
v-IH
restrict to
TITs+I + T2 Ts+2 +
T 2
+
+
n
+ TsT2s
if
p '" 2
or if
if
H of degree
is defined by a
P '" 2 , and
n = 2s
or if
semi -ordinary, and
y'
is
Part (iii) of (22) is
in the T's whose initial form is the quadratic form, 2
p = 2 , the smooth sur
is a hypersurface; on the
v, the
correspond bijectively to the hypersurfaces
there exists an
polynomial of degree
[1956J, which asserts that iff a general central proj result implies the other. directly.
n
2s+1
and
n-!
if
p
Probably, more than i
2
reasonable to conjecture By (II), n is odd.
hep,H) = n
if
is even, and
=
p = 2
and
Hence, (i) results from (17), and (ii) results from (18) and (19).
Similar, if more involved, reasoning yields the next two theorems. (21)
THEOREM (Hefez-Kleiman [1984], (5.6».
a hypersurface of degree> 2 , or let
> 2.
If the characteristic
is ordinary. (22)
If
p
2
Let
Y be the section of
p '" 2 , or if the dimension
and if
n
is odd, then
Y
of
X by a general hyperplane.
if
X
is ordinary, then If
P = 2
and
Y X
Then:
n
X by
is odd, then
Y
is semi-ordinary. Let
Y be the section
(i) If the characteristic
p '" 2 , and
is also. is ordinary, then
Y
is semi-ordinary and
y' is
a hypersurface. (iii)
If
X'
is not a hypersurface (in other words,
case for all other
n' '" N-I ). then
t
p, a
unique point of contact, cy
Y be a general hypersurface of degree
THEOREM (Hefez-Kleiman [1984], (5.9), (5.12).
(ii)
1
It is easy to verify
if Indeed, fix
Indeed
true, except that, in (iii
vX
is semi-ordinary.
pN
is odd.
the 3-fold
(ii) (Hefez-Kleiman [1984], (5.4»
of
n
In (22),
is ordinary.
d
is ordina:
y
hypersurface that works fo: even, then
X is neither ordinary nor semi-
Then: (i) (Katz [1973], Thm. 2.5)
Tl' ... , TN
L
A number a f remarks m,
a
> 2.
X
is semi-.
X
Kleiman [1984J, (5.10), (5
ordinary, then this condition is a quirk of the embedding.
d
If
(v)
a matrix of this form
X is ordinary, except either if
Consequently,
The next result indicates that if
(20)
If
The Hessian matrices are all those of a certain special form,
and the problem is to construct, for each if
and
0'+1
(iv)
-> y'
is probably pure
this conjecture is true, t work also if (22a)
p
2 •
THEOREM (Hefez-Klei
by a general hyperplane
M
X.
t
If
X
is ordinary,
the cone of tangent lines ordinary and
p '" 2 , then
is not the dual of a reduc together imply that
X is
The first assertion straightforward. First of
TANGENCY :onjecture.
dimey')
= (1,2),
[(d,e)
2S
Thorup
(iv)
.ch case with the aid
X Y
and
e
X is semi-ordinary.
2
and
n
Y
a
se embedding for some
IPN- J
oder
vX
v , the H
0
f degree
is defined by a
p
and
y'
n.
Hence, if
p
=
2
and
n
p-
is a is
Y may also be viewed as a subvariety of the cutting hyperplane, dim(Y')
X: w
X'
xy+yz+zx
if P
=
= n' .
= x 2+yp+J
X; z
if
p
3
(resp.
2) is (irreducible and) semi-ordinary and
is a hypersurface; by (22)(iv), then
Y
is ordinary.
Hence, (22)(i)
is not reversible, and (iii) is sharp. if p = 2 , the smooth surface X: w3
Also (ii) is not reversible. Indeed, = x 3 +y3+ z3 is not reflexive, and X'
is a hypersurface; on the other hand,
y'
y'
semi-ordinary, and
is not ordinary by (18), so it is
is a surface because
Y
is a curve but not a line.
Part (iii) of (22) is dual to a result, Lemma 3, p. 334, of Wallace [1956], which asserts that, if
X
is not a hypersurface, then
iff a general central projection of
s the quadratic form,
is a hypersurface also if
y'
It is easy to verify that the surface that
restrict to
:ur faces
In (21),
true, except that, in (iii), now
vX
ogeneous coordinates
... ,
is ordinary .
X is ordinary or semi-ordinary.
In this way, the conclusions take on a new meaning, and they remain
the 3-fold is odd, then
is also.
is in fact not reflexive.
In (22),
n
Y
181
Indeed, Ein [1984], II, (1.3) gave a proof that
hypersurface that works for any
g.
is even, then
Y
is semi-ordinary, then is ordinary, then
is odd.
even, then
iinary nor semi-
n
is reflexive iff
Kleiman [1984], (5.10), (5.11).
+I
d = I
If
X
DUALITY
A number of remarks may be made about the above two theorems (see Hefez-
a matrix of this form i ther if
If
(v)
:ertain special form, is odd, resp.
n '+ 1 ,and
AND
result implies the other.
X
is reflexive.
X
is reflexive
It seems that neither
However, it is not hard to prove Wallace's result
directly.
= 25+1 =
0-1
and if
p
=
Probably, more than is stated in (22)(ii) is true.
2
p = 2
and
s from (18) and (19).
t
two theorems.
8
the section of
CY -> y' X
by
is odd, then
Y
,emi-ordinary.
,t
p ,;. 2 , and
emi-ordinary and
;,
y' is
Of';' N-I ), then
2
and
X
Indeed, it is
is ordinary, then, as is the
p, a general tangent hyperplane
H
to
Y
is tangent at a
P. Moreover, YH is probably of length 2; that is, is probably purely inseparable of degree 2. If the first part of
this conjecture is true, then the proof of part (ii) of the next theorem will work also if (22a)
p
=
2 .
THEOREM (Hefez-Kleiman [J984], (4.10».
by a general hyperplane
Y be the section
:teristic
case for all other
p
unique poir:t of contact,
ypersurface of degree n
reasonable to conjecture that, if
X.
If
X
M, and let
p
X'
Y be the section of Lemma d, p. 5) (i)
drawn from the point
M'
2 • then (ii) the hyperplane tangent to
is not the dual of a reduced component of together imply that
X
X
H be a general tangent hyperplane of
is ordinary, then (cf. Wallace [1958],
the cone of tangent lines of ordinary and
Let
X 0 H.
If X'
X
is
y'
is
at the point
H'
Conversely, (i) and (ii)
is ordinary.
The first assertion is well known, and its proof is more or less straightforward. First of all, it is not hard to see that, whether or not
X
STEVEN L. KLEIMAN
182 is reflexive, the preimage of then so is
is the cone of lines from
y'
Y
in
ex.
M'
Hence, in any case, if
X'
Z meets
Moreover, it is clear that
y' .
suffices to observe that, for every point
Z in
to the image
of
is a hypersurface, X,sm Hence, it
H'
(B) The point
(P,H)
(e) The point
of
X,sm
lies in
= ex
ex'
of
(E) The line from
p'
lies in the pre image of
to
H'
Y .
M
H tangent to
hyperplane
at
X'
H' .
= TH,X'
p'
since
X'
consisting of a point
H-contact locus
and
of
Q
hyperplane tangent to
X'
at
H'
(X 0 H)sm.
is tangent to
H
Then,
hyperplane
s
reduced component P.
Hence,
at
X
Z of
n = 1
Howe
X 0 H , a co
In (22a), assume ordinary.
P
tangent to
p'
component of
Moreover, this hyperplane contains G'
.
p
automatically reflexive.
X, a
Q corresponds to the
is ordinary,
X
Hence
P.
In (22a), assume
(X 0 G)H ' and (3) the G-contact locus of each reduced
Q lies in
because
p' .. TH,X'
because, by the above argu
X at
component of eX Q H) , if nonempty, is contained in
fr
L
be a general poin
G'
is a
Q, and a hyperplane G passing through Q is transverse to Xsm (2) Q is the entire support of
is such that (I) G the
(Q,H,G)
SO,
Hen·
contradiction to (4).
hypersurface. The proof of the second assertion is more involved, and it is based on showing that a general triple
lies in
simple point
M'
is tangent to
H'
Q' ; H.
Now, the line
Let
contains the point
M'
(l
whose points represer
XG
Conditions CD) and (E) are equivalent because
X
the dual of any component
Z .
lies in the hyperplane
P
CD) The hyperplane
point of component of
*
following conditions is equivalent to the next: CA) The point
M does not contain any po
IP N
eX,sm , each of the
of
(P,H)
X'
p >
If it did, then
However, this point does not lie in the dual of any reduced component of
not imply (ii) if
eX Q H).
proved, using arguments si
P
If it did, then
by (3).
It would then follow from (I) that
=Q
(X Q H)
G would be tangent to
P
Therefore,
P
by (2).
P
and
Q is not.
Thus, (ii) holds.
H
at a simple point
is tangent to
eX
p # 2
G)
at
(X 0 H)
is a simple point of
The hypothesis
Q
is illustrated with the va 1-7. SMOOTH VARIETIES.
To prove the converse, note first that Hence,
y'
X'
}j'
contradicts the assumption that
is ramified at every point of
M'
M'
would be empty, and
n
= O.
y'
X'
In view of these two notes and of
XH
is finite, (2) the hyperplane tangent to
the hyperplane
Q' , is not the dual of a component of
given a general hyperplane through H'
is equal to XH .
at
(X 0 H)
H'
is such
X
, and (3)
Q , the cone of tangent lines to
X
is reflexive, it obviously suffices to prove that
Suppose not.
Let
that, if
M be a general hyperplane through
Q.
n'
N-J.
from
Q Then
n > 2 , then
c
Another unexpected r' (23)
THEOREM (Zak's tang'
nonlinear, then
, which is
y'
To prove that lies in
X'
CX,sm
Cn+!
c
a consequence of (24) belt
X' , which
of
2 ,then
n
of tangent lines to (Q,H)
1
n .
Zak's result is a ca The idea is simple; see [198IJ, 4.3
L
Pick point
does not lie on
X.
Consider the set of pairs
TANGENCY he image
X'
in
Z
X'
of
is a hypersurface, X,sm
Hence, it
F N*
Y .
I 0 H.
X 0 G
lies in
G'
-x'
point
is a
X'
In (22a), assume
of
Q
Q
locus of each reduced (] rOsm.
Then,
hyperplane
p'
at a simple point (X 0 G)
at
(X 0 H)
2 enters in
P.
Hence,
P
Z
n
of
Indeed, if
positive dimension. H' • So (i) would
X'
X'
these two notes and of at
V n y'
is nonempty.
G is tangent to
Y at a at
X 0 G
CX,sm
is such
alone implies that
X
P, in
X'
X
at
is reflexive,
H'
However, then
P
Z
is
Indeed, say that is the dual of a
p'
is reflexive as
p
=0
, then
Z
supports an isolated, nonreduced Z
is reduced.
Then (i) alone does not imply that
If it did, then (i) would imply (ii) by (22a). p> O.
X
represents the
P
Suppose that
Since
p > 2.
is ordinary,
is a hypersurface, and because
X'
Since
X
is
However, (i) does
In fact, in Hefez-Kleiman [1984], (4.11), it is
proved, using arguments similar to the preceding ones, that if (ii) fails and if
X' is a hypersurface of degree> 2 , then (i) holds.
is illustrated with the variety,
X: T p+1 +
o
Lastly, assume that
X
Moreover, this result
=0
+ T p+l N
•
is smooth and nonlinear.
Then
there are some unexpected restrictions on the dimension equivalently, on 2 , then
z
n'
c
z
(n+n')-(N-I) .
= N-l.
If
a consequence of (24) below.
Note next that
x'
H' , because
M is tangent to
X Q H , a contradiction because
not imply (ii) if
n
X' , which
tangent lines to of
So
y'
Then
(i)
X0 H
1-7. SMOOTH VARIETIES.
ypersurface.
Then
O.
z
tangent to
reduced component
ordinary.
uced component of
,H)
lies in
V Q y' .
ln
H'
at
It
Moreover, (ii) is easier to prove.
X at
is tangent to
In (22a), assume
erplane contains G'
Ie point of
H
component of
sponds to the
angent to
p
automatically reflexive.
the entire support of
of
L
V of
does not lie in
M does not contain any point of
is tangent to
(5) implies that
because, by the above argument,
X , a
G passing through
o H)
Hence
P.
H'
M'
contradiction to (4).
since
I, and it is based on
l
G such that (4)
and (5)
to
Hence,
be a general point of
simple point H' .
from
L
= TH,X'
pI
M' is a
Moreover,
Hence, there exist s an open neighborhood
M'
Now, the line
183
does not lie in the dual of any
whose points represent hyperplanes
the dual of any component of
Let at
M'
XG M'
DUALITY
XH because of (I).
Q' ; so, by (2),
general point of component of
CX,sm , each of the
reimage of
M does not contain any point of
AND
that, if
n > 2 , then
X
n ' of X' , or For example, 11-(21) asserts this: if
is reflexive too, then this assertion is also
Now, it follows by taking hyperplane sections
c > 2
Another unexpected result is the following theorem. (23)
THEOREM (Zak's tangency theorem, Zak [1983]).
nonlinear, then
If
X
is smooth and
n' > n .
H' , which is Zak's result is a consequence of the Fulton-Hansen connectedness theorem.
(X Q H) , and (3) ent lines to
X
from
The idea is simple; see Fulton-Lazarafeld [1980], Cor. 7.4, or fujita-Roberts [ J 98 I ],
=s to prove that lane through
Q .
Q Then
L
4.3
Pick points
does not lie on
I.
Consider the set of pairs
QO in XH and RO in Choose a point P in L-X. (Q,R)
where Q
X-H Then
is a point of
such that their join P XH
is not on and
R
H .
is one of
STEV"!':N L. K IEHAN
184 X such that
R, Q
and
and its complement. Suppose
n.
n'
P
are col inear.
SO
XH and its complement. H , a contradiction. (Q,Q)
in
P.
of the projection from
common to
P
lies in
TQX ,
THEOREM (Landman's parity theorem, Landman [1976J).
reflexive but not ordinary nor linear, then
If
presented in the next section, after II-(22). of
c
X
s smooth and
recent
c/2 , and it does not require the full strength of smoothness. HOwever, it is still necessary that Xsm be large enough to contain the contact locus of a
is represented by the
ones from de formation th the Belinson spectral se and the adjunction mappi Independently, Holme and
is even.
THEOREM (Ein [1984
(27)
(25)
THEOREM (Ein's bundle theorem, Ein [1984J, II, 2. I).
smooth and reflexive but not ordinary nor linear. C
CX Q Z
Let
N(C/Z)
denote the normal bundle.
Assume that
Z = XsmxX,sm
Set
X
is
and
(26)
Then there exists a
Hom ( N(C/Z),
reflexive but not ordinary nor linear. X' , such that
X
H be a hyperplane, corresponding
XH is contained in
Xsm.
Moreover, if
Then there
N(XH/X)
is of rank
XH The c , and so
c I ( NeXH/X) ) As to (25), let I Z.
s
Then
s
a section of
s;
If
n
X is
3
is the Grassmannian of lines in
lP4
X
6
and
is a scroll; moreover, if
nt < N-2 , then either X
n-(c/2)J-planes, or
X
embedded in
by the Plucker map, or
X
is ruled by
is a hyperplane section of this
Grassmannian. THEOREM (Ein [1984], 11-4.4). (b) JP
then
c
lP S
If
X is ruled by nonintersecting [n-Cc/2)]-planes.
n
but not linear. LZ,
and
Then: (i)
nonintersecting
(28) ) )
embedded in
n, then
(iii)
with
In a given basis,
s .)
ones from deformation theory and the theory of uniform vector bundles, the
I)
A
s(x+y)-s(x)-s(y)
(Recall
defines a
Lefschetz-Barth-Larsen theorem, Fujita's classification of Del Pezza manifolds,
and
Xsm.
=
s
Starting from (25) and using a variety of methods and results, including
unless
Then there exists a
in
b(x,y)
is represented by the matrix of second partials of
(27)
Assume that
by the rule,
Hence
Independently, Holme and Schneider proved (27)(i).
"en. . I).
b
is an isomorphism ln view of (I I) and (12).
C
is the
The map is an isomorphism because its fiber
ext result. If
N(C/Z)
If 1
Assume that
X
is smooth and reflexive
n = n' 2N/3, then X is either: Ca) a hypersurface, embedded in lP 2n - 1 by the Segre embedding,
ec) the Grassmannian of lines in
embedded in
JP9
by the PlUcker
map, or lP 15 .
is simply the
Cd) the IO-dimensional spinor variety in
ion of (26) follows,
Here are some remarks about (28) (they are, for the most part, taken from Ein's two articles). n > 2N/3, then
c
IPNxIP N*
that defines
the ideal of
C
in
'act, the restriction is
,rplane
H, restrict
X
First, according to Hartshorne's famous conjecture, if
is a complete intersection.
complete intersection, then
X'
Now, if
X
is a (nonlinear}
is a hypersurface by 11-(12)(ii).
conjecture implies that the list contains every smooth, reflexive n = n' .
Now, in every case but Ca),
embedded variety, smooth.
X'
In case (a),
is isomorphic to X
X
Hence, the X such that
is self-dual in the sense that, as an X
is self-dual iff
in particular, X
is a
X' iff
isalso
x' is smooth;
STEVEN L. KLEUlAN
186 see II-(IO).
Note finally that, by virtue of (24), if
X' is smooth, then
reflexive and if
X
is smooth and
of the term "polar" in
n = n' .
paragraphs, some of the details, see Fulton [19!
THE
II.
II-I.
RANKS
Consider the Gauss
The setting will be as it was in the bulk of Section I.
Namely,
X will be an n-dimensional (reduced and irreducible) subvariety of
the projective space
JPN
p,
and
characteristic
over an algebraically closed
is simple on
fir s t assert ion follows f-
by the vanishing of the maximal nlinors of the augmented
(i+2)th exterior power of the map, hypersurface and
Moreover, if P
because
is the Jacobian locus of the system of hyperplanes
equations defining
obvious.
The scheme structure is
Jacobian matrix associated to a regular system of parameters at of
P
need not be equal.)
This time the scheme structure is most naturally defined locally P
theorem,
(TpX)
JPi+ 1
that is, it is the locus of singular points of these
at a simple point
Y=X0
Indeed, set
general plane of codimens
here most naturally defined by the appropriate Fitting ideal. containing
i (X)
>0 )
( e
q
closure of the ramification scheme of the central projection from (X sm _ A)
is ev
(4)
and that of t
the case of plane curves
is reduced
XCA) , which are valid in arbitrary characteristic.
Xsm
,
P .
not be used below and will only be discussed briefly.
Note that
(x'
starting from Severi's d
There are some other interesting and well-known geometric interpretations of the scheme
f
r N- 1- i (X')
from the general transversality results (Kleiman
[1974] and Vainsencher [1978J, (7.2)) that the interpretation of the
(4) r i (X)
discovered early in the
n ,then
X is reflexive.
° , it follows
is reflexivE
related, as follows.
ex =
Indeed, by the
R -> X(A) , restricts to an isomorphism over
above, the map,
X,sm
A.
x
If
i > n-c (that is, n' < N-
is smooth and, if
l-i ), irreducible; moreover, its preimage is dense in assertions about
= ex'
ex
CX -> X' , is a bundle of projective spaces over
Moreover, the map,
respect to
are each counted with
r·1
X(A)
with the hypersurface with equation,
a 1F 1 + ... +N-IFN-l = where F.1 is the ith partial derivative of F This hypersurface is traditionally called the polar hypersurface of X with
second assertion.
MoreovE
Piene gave somewhat diffeI v : Vx
->
pic 1)
.
If
X
It is evident, that
TANGENCY y.
respect to are each counted with
if the characteristic
I,
,e::1erically reduced, and CX = ex f ,ve spaces over X,sm
,ypothes is,
> n-c (that is,
nf
,ow.
(4)
< N-
A .
X
is reflexive, then its ranks and those of its dual
rN_I_i(X') Indeed,
(4)
for all
X'
are nicely
If
X
is reflexive, then
i.
is evident in view of the the definition of reflexivity,
ex' , and that of the
CX which is
189
THEOREM (Piene [1978], 3.6; Urabe [1981], 3.3).
r i ' (I).
On the other hand, (4) is not evident
from Severi's definition of the ri ' (3), as Piene and Urabe did.
Indeed, by the
.sm over
DUALITY
related, as follows.
ri(x)
Thus, the
R
If
AND
Perhaps, this is the reason why (4) was not discovered earlier.
Of course, in
len-Macaulay by the
the case of plane curves and in that of curves and surfaces in 3-space, (4) was
!.
discovered early in the 19th century. then
X(A)
that, if
The next result is a useful comparison theorem.
is reduced
i = n-c , then
THEOREN (Piene (1978], (4.1) and (4.2».
(5)
P
If
in the above
Q is a general point and
(e > 0 )
q = pe
Indeed, set
lracteristic.
These will
is the
X(A)
jeet ion from
P
obvious.
Moreover, if
o
A.
i
a
I,
N-1
is sirr.ple on P
Y
if
for
N-2
i = 0,
is simple on
P
is simple, then i+1
Q , then
the projection map from
Y = X Q M and consider a point of
theorem,
general plane of codimenslon
A
q
ri(qX)
r i (X)
interpretations First,
for
r
!rsality results (Kleiman
M is a general hyperplane,
If
TpY
Y
(TpX) Q M.
B Q M
such that
By Bertini's
X, and the converse is
(Notice, however, that the hyperplanes
Let
Then
A
TpY + Band
B be a
(TpY) n B TpX + A
need not be equal.) The points of
The scheme structure is
s
because
ideal.
X(A) 0 M that are simple on
M is general.
Hence
Y(B)
for example, if
points of these
So, the
are reduced,
In general, the scheme structures
of
imeters at
statment about the Schubert varieties is easy to check. In the first assertion,
ime globally, by the
case that A
X
A(aO'
must be considered.
Y(B)
by those on the corresponding Schubert varieties, and the corresponding
M
introduced above.
Y(B)
Yare reflexive.
and
minors of the augmented and a basis
and
and
X(A)
laturally defined locally P
X(A)
X
form a dense subset,
X(A) n M , at least as sets.
first assertion follows from (3), at least if
Item of hyperplanes
X
,a
TI'N-I
However, these structures are induced
X 0 M could also be viewed as a subvariety of
If it is, the assertion rema ins valid.
is a
to prove similarly that the ranks of any subvariety of
, aN)
and in TI'N.
it is clear that
X(A)
Indeed, it is not hard
M are the same in
second assertion.
Noreover, the second assertion too may proved similarly.
face with equation,
Piene gave somewhat different proofs, which deal more formally with the map,
1 derivative of
v : Vx -> piC I).
ypersurface of
F X with
M
On the other hand, this fact is an immediate consequence of the
If
X
is smooth, then (5) can also be proved using (8a).
It is evident, that the first assertion of (5) is equivalent to the
STEVEN L. KLEIMAN
190 I0l10wing lovely
,.,. tr
is a certain amount
of the ranks.
wt .. (
completed Poncelet's THEOREM (C. Seg!"e []912], p. 924).
(0)
Xi
class of the section
X by
of
The ith rank of
is equal to the
X
general hyperplanes; in other words,
ri CX )
THEOREM (Hefez-Kleiman [1984], (4.13».
The special case of a
r.
ri
n-c < i
iff
0
Because of (2), it remains to prove that
Proceeding by induction on
n, assume that
general hyperplane section
Y
then
ri
then
y'
r.
0
if
First, if
r·1
If the characteristic P '"
n-c < i < n .
n-c(Y) < i < n
if X'
cCY)
= n-i
; hence, (7) holds in this case.
is not a hypersurface, then
c(Y) '" c
SMOOTH VARIETIES.
otherwise indicated.
Second, if
is not ref:
11-3,
assume X is smooth, unless N(X/W N) (the dual of the ideal
modulo its square) is locally free everywhere on
X
To prove (i), note that
any rate, if Suppose
-I
is necessary so that the tautological class of the right hand
side is equal to
h ; the twist arises technically in the identification of the
graph of the incidence correspondence I
with the cotangent bundle of
W
are now defined everywhere on
obviously,
X.
[X]
If
X is a hypersurface, then (8a) yields the following expression for the The case of a plane curve was given by Goudin and du Sejour (1756). PROPOSITION.
If
X
is a smooth hypersurface of degree
r·l
d(d-I)n-i
Indeed, here, obviously,
for
N(X/W N)
i
is of degree 2
X'
0,. .•
d , then
P
=2
and
n
is,
ex -) X ex
is a hyperplane and
d
rn(X)
So, by (2)(iv),
X is a hy
(9),
d ' '" d(d-J)n
There
It is interesting to n
Hence,
15a), if
X
characteristic
is a smooth p
divides
if also the dimension
n '"
nice to know what happens w
1+ (d-l)h + (d_I)2 h 2 + •••
For a smooth complete
Therefore, (8a) yields (9). Poncelet observed that, if
Thus, X'
To prove (ii), note tho
, N- I
I/[ 1 - (d- J)h]
s( N(XIWN)(-I) )
Xis re flexi v(
is a hypersurface, X'
1-(
0Xed).
or if
is ordinarj
by (5) and (9),
ri
(9)
is
hypersurface, say of degree
}hiSn_i(N(X/wNH-I»
ri
If
In terms of them, the ranks are,
given by the following expression:
(8a)
p'" 2 X
linear subspace.
N
The Segre classes (better, operators), or inverse Chern classes, of N(X/W N)
CX -> X'
cone, say with vertex V • ; However, if
The twist by
X' is
therefore. X' is a hypersuri
whence,
There fore,
w( N(X/WN)(-l»
CX
is also
hypersurface) and if
p i< 2 ,then Here and in
X
Conversely, if
(il)
X'
by 1-(22)(iii), or by (5) and (2)(iv).
Then, the normal sheaf
(8)
X'
is odd, then
purely inseparable of degret
Therefore, (7) is valid. 11-2.
is ordinary and
is a hypersurface,
is a hyper surface (see the first paragraph after the statement of
1-(23», and so
Supp
(i)
n
and that (7) is valid for a
By (5), therefore,
Now, there are two cases.
0
ri
n) 2
COROLLARY.
(10)
The proof of (7) is so simple, it is a wonder that (7) was not found earlier.
[1978); a review of Pohl's
ro(X i )
The next result, a nonvanishing theorem, nicely complements (2).
(7)
carried further by Pohl (19
X
is a singular plane curve,
its class
a little more complicated,
is a certain amount less than
"".
X is equal to the
'f
anes; in other words,
completed Poncelet's we"
d(d-l) •
(1839) corrected and
(see Piene [1978J, Rem., p. 268).
[1978]; a review of Pohl's and
(10)
n-c
iff
I , then
for
i
>a
sn > 0 , so
For (ii) observe that rO > 0 , so
X'
sn
(dl-I)n ; hence,
is a hypersurface by (2)(iv).
Like considerations app
Of course, (i) is also an immediate consequence of (2)(i)(ii). Whether or not
X
torial formula for the classes (13)
Cj(X)
is a hypersurface, (8) leads to the following combinar·1
In terms of the hyperplane class
For all i ,
2:
r·1
])j(I!I)
j;i
Indeed, the tangent sheaf--normal sheaf sequence of N(X!W N )( -])
in
K(X) •
h
and the Chern
(these are the Chern classes of the tangent bundle
PROPOSITION.
X
TWN
COROLLARY.
( 16)
If
X
The following case
hj
in
sists of deg(X) points,
Tx ). (X)
WN
•
yields
geometry. ( I 7)
(sometimes called the
then
Euler sequence) now yields
It was state
COROLLARY
plane curve
(TWN/X)(-J) - Txe I)
Using the standard presentation of
X is a curve,
If
E
ro
(The E of degI
e ( e + 2J
To derive (17) frl
Nex/wN)(-I)
(The latter is a well-\
- (OX + TX) IllOx(-I)
method of "adjunction" Hence, (13) results from (8) and the following standard expression for the Chern classes of the tensor product of a bundle
E of rank
e
and one
L of
rank I (Fulton [1984], Rem. 3.2.3, p. 55): C
r
(Elat)
2:. (e-:t;+j)c J
J
c(X) = cO(X) + c1(X) + •..
The result is this.
g
as the ,
the structure sheaf of Since
.CE)c (1)i r-l I
For i = 0, (13) may be put directly into a more compact form involving the total Chern class
by viewing
raCE)" e(1
( 17a) Now, formula (17a) is Kleiman [1976J, p. 365
TANGENCY ds (11) below.
(14)
This
ny applications of
DUALITY
193
COROLLARY (Katz [1973], (5.6.1), p. 39).
11-3.
mplished before the
AND
TOPOLOGY.
expression for
In turn, (14) yields a lovely ar.d useful topological rO'
It involves the topological Euler characteristic
E(X) ,
which enters through the Gauss-Bonnet formula (see Griffiths-Harris [1978], p. 416; Fulton [1984], EX. 18.3.7(c), p. 362; Kleiman [1968], 11-24, p. 322):
ntersection of e the coefficient of
ti
E(X) (15)
THEOREM (Katz [1973],(5,7.2), p. 250).
section of
he end of the last
i
one of
XI ' and
X
Let
XI
be a general hyperplane
Then
To derive (!5) from (14), observe that
N-I
0, .• "
Now, because on)
Xl
in
XI
represents
h , the tangent sheaf--normal sheaf sequence of
X yields
not an n-plane, then
[c(X)/C l+h)]/X 1 Hence, by the projection formula and the Gauss-Bonnet theorem,
the d's are 1, then cCX)h/( l+h)
sn > (dj-l)n ; hence, ypersurface by (2)(iv).
i)(ii).
If
the following combinaclass
h
and the Chern
gent bundle
(i!l) f
X
Like considerations apply to the inclusion of
Tx
).
pN
COROLLARY.
geometry.
yields
If
is a smooth curve of genus
X
g, then
(17) then
sometimes called the
COROLLARY
(The Bischoff-Steiner Formula),
E of degree
ro
e
If
X
method of expression for the L
by viewing of
d
= ef
and that
g
2g-2
e(e-3) ,
It can be proved by the
ion" used above to obtain the formula for
E(X 1)
above, or
as the arithmetic genus and relating the Euler characteristic of
the structure sheaf of Since
is obtained from a
e ( e + 2f - 3 )
To derive (17) from (16), note that
and one
= 2g-2+2d
by reembedding it via the f-fold Veronese map,
(The latter is a well-known formula of Clebsch (1864).
e
rO
It was stated by Steiner (1854) and proved by Bischoff (1959).
plane curve
1)
rank
' and (15) follows.
Xl
is a curve, then
The following case of (16) is of historical importance in enumerative
hjCn_j(X) . in
in
X2
X2 is empty, whence E(X 2 ) a, and Xl consists of degeX) points, whence E(X 1) = deg(X). So (IS) has this corollary. (16)
X
raCE)
X to that of
= e(e-I)
),
by (9), the formula in (17) may be rewritten as
( 17a)
'mpact form involving the suit is this.
Now, formula (17a) is valid also when
E
is singular; one proof is given in
Kleiman [1976], p. 365, and another is given in
§
III-3.
STEVEN L. KLEIMAN
194
In the setup of (17), the points of of degree
f , and the points of
X'
Suppose that the characteristic is e
> 2 and E is general.
Hence, by 1-( 17), a general and
i(P, E.F)
degree of degree
=
X'
f
2.
F N*
represent the plane curves
represent those p # 2
F
tangent to
Then, by I-(20)(i) or 1-(21), F
f > 2
and either that
tangent to
E
X
F
E .
or
formal propert ies.
that
rO
Lane
which are not in print,
Here is the key rei
is ordinary.
is tangent at a unique point
Moreover, by (Z)(iv) and I-(15a),
vector spaces over a fiE
P
is equal to the
(18)
LEMMA (Landman [1'
general hyperplanes.
Thus, for example, the degree of the hyper surface of curves of
2 tangent to a line is
tangent to a conic is
2(2f-1).
of the degree of the hypersurface
but of the degree of the relation among
the coeffecients of the equations of the curves number such of
F
F
tangent to
n
Indeed, because of
Of course, in the last century, they spoke not X'
(b -
rO
f >
2f-2, and that of curves of degree
table of Betti numbers
E , or of the
X
bn -
Xl
b n-
X2
bn-
in a general linear pencil.
Steiner (1848) implicitly and Bischoff (1859) explicitly went on to conclude from Bezout's theorem that there are 6 5 = 7776 conics tangent to 5 others in general position.
This number is wrong!
It is wrong, because the 5
hypersurfaces of conics do not intersect in a finite number of points; in fact, each contains the Veronese surface of double lines.
Before Cremona (1964)
published essentially this correct explanation, it had been discovered that the method gave incorrect results. Indeed, it gives similarly 25 = 32 for the number of conics tangent to 5 lines; however, this number was known to be
It is now evident that
Euler characteristic is numbers.
Finally, each
Theorem, and the second
The next result is (18) reduces to (16).
The actual number of conics tangent to 5 others is the number of points of intersection of the 5 hypersurfaces, residual to the Veronese surface; so the number may be obtained from a residual intersection formula.
Severi (1902)
(19)
is even.
Indeed, (19) resu;
proved such a formula and obtained the correct number; for a modern treatment of this point of view, see Fulton [1984], Ex. 9.19, p. 158.
On the other hand,
the correct number, 3264, was first published by Chasles (1864), who obtained
THEOREM (Landman
"odd Betti numbers are the bilinear form on
it from a rather different point of view, one more symmetric with respect to
is the hyperplane dasl
the duality; see III.
Poincare duality; if
(However, Cremona (1862) published a formula which gave
it; Berner (Diss., Berlin, 1865) obtained it by a method of degeneration; and de Jonquieres (1866) claimed to have known it before 1864.
For a more complete
historical introduction to Chasles's theory, see Kleiman [1980J.)
Here is, perhaps, (20)
THEOREM (Marchio
codimension at least 3
Landman [1976J, starting from (15) and applying Picard-Lefschetz theory, obtained some remarkable results. bi
= bi(X)
The results involve the Betti numbers,
Indeed, applying
Then the third term in
, namely, the dimensions of the cohomology groups,
Virtually this pr b.1
dim Hi(X)
from a version of (15)
The type of cohomology theory is secondary, provided it is a "Weil" cohomology
in Lefschetz's L'Analy
theory; see Kleiman [1968').
rO = rZ-l
The groups may be simplicial if the ground field
is the field of comlex numbers, de Rham or Hodge if the characteristic etale if
p
is arbitrary, or crystalline if
p > 0 , etc.
p = 0,
The groups must be
iff
X is
[1957J found that values of
ro
ro
can occ
TANGENCY ,nt the plane curves F
tangent to
that
f > 2
I),
F
formal properties.
that P
is equal to the of curves of
(18)
Then
Let
rO
X by
denote the section of
i
is the sum of 3 nonnegative terms:
f >
=e of the relation among 1gent to
LEMMA (Landman [1976J).
general hyperplanes.
century, they spoke not
t
Landman worked over the complex numbers, but his proofs,
Here is the key result.
ent at a unique point
curves of degree
195
which are not in print, are simple and formal.
X is ordinary. rO
DUALITY
vector spaces over a field of characteristic 0, and possess the appropriate
E .
or
AND
Indeed, because of the Weak Lefschetz Theorem and of Poincare duality, the table of Betti numbers is as follows:
E , or of the
x
licitly went on to conics tangent to 5
16
is wrong, because the 5 llllber of points; in fact, ,fore Cremona (1964) been discovered that the 2 5 = 32
Irly
for the
,er was known to be : the number of points of
It is now evident that the formula in question results from (15), because the Euler characteristic is, by definition, the alternating sum of the Betti numbers.
Theorem, and the second and third, by the Weak Lefschetz Theorem.
(19)
'mula.
is even.
for a modern treatment
158.
On the other hand,
s (1864), who obtained
with respect to ed a formula which gave
64.
the bilinear form on
n
is odd (and
Poincare duality; if
rO
Hi(X)
n ) sending
(i
The latter is true because (x,y) to xyh 2n - i ,where
h
i
is odd, then it is skew-symmetric, so
bi
is even.
Here is, perhaps, an unexpected result. THEOREM (Marchionna [1955]).
If the singular locus of
>
codimension at least 3, then
r
n
Then the third term in (18) is just
rn-I
Virtually this proof in the case
n
as
=
X
is of
-
Indeed, applying (5) repeatedly, we may assume
the Betti
roups,
X is smooth), then
is the hyperplane class, is nondegenerate by the Strong Lefschetz Theorem and
(20)
card-Lefschetz theory,
If
"odd Betti numbers are even" (Landman's slogan).
For a more complete
[1980].)
THEOREM (Landman [1976J).
Indeed, (19) results immediately from (18) and from the general fact that
d of degeneration; and
n
However, it is consistent with (16); in fact,
The next result is curious.
(18) reduces to (16).
,ronese surface; so the Severi (1902)
Finally, each term is nonnegative; the first, by the Strong Lefschetz
2
n
X
is a smooth surface.
2, and (20) follows.
was given by Marchionna; he began
from a version of (15), which he called the Picard-Alexander formula and cited is a "Weil" cohomology
,1 if the ground field
characteristic
:c.
p
=
0,
The groups mus t be
in Lefschetz' s ro
=
rZ-1
iff
situs ... (1924). X
[1957] found that values of
Marchionna went on to show that
is the plane or the Veronese surface.
can occur.
iff
X
Gallarati [1956J,
is ruled, and that, when
rO > r2 ' not all
Lanteri [1984] carried Gallarati's work further.
STEVEN L. KLEIMAN
196
check.
Coupled, (20) and (2)(iv) immediately yield this: (21)
THEOREM (Landman (1976]).
then
X·
If
X
is a smooth surface but not a plane,
is even.
is a hypersurface.
Assume that
Namely, Lascoux [
from which he could show So, by (2)(iv),
Now, it is clear from
X is reflexive.
(21).
Then there are two other proofs of
c by (2)(i)(ii).
One is a more or less straightforward approach, due to Piene (pvt. comm.,
(4); hence, if
1977).
(24); whence,
The second proof, which is due to Landman [1976J, is based on the
parity theorem, 1-(24). but
Namely, the latter asserts that, if
X' is not a hypersurface, then
< n = 2 , therefore either X
=n
c
the codefect
=2
and
c
X
is even.
is reflexive Since
X' is a hypersurface or
Q defines
is isolated in
P
intersection takes place in the hyperplane
ex
is, by definition, the fiber of VH ;.
is that of
CV
Finally,
I , then it is transvers
is isolated in
ex
0
is isolated in
ev.
point at
P.
ex
SUppOSF X 0 V
the
holds because
H , viewed as a point of
the intersection
X and
V at
0 CV , and if
The latter condition
is
X 0 V
X'
XH
If
(P,H)
P ,because
(P,H)
if it is proper, if
X 0 V has a nondegenerate double P
is
TpX 0 TpV , by a quadratic form whose
dim(X) + dim(V)
'" N-I , then the tangent
If the third condition is satisfied, then
N
is a scheme of length 2, or equivalently,
if the characteristic
p
#
i(P, X.V)
2.
is nondl
TpX 0 '
Conversely,
Conversely, if tl
tangency is simple, then The preceding proof Katz's proof of I-(Jn, a observe how 1-( I
n
is a
CI
X.
rn
hyperplane
H of
Suppose
is general.
H
hypersurface.
If
(P,R)
nondegenerate double CH
meet in
point
by (I)(iii), and so the third condition is trivially satisfied.
dim(X) + dimeV)
II(p,H)'
IICP,H)
quadratic form, then
, resp.
f dimension
that the tangent cone at
defined in the Zariski tangent space,
0
VH ' and because the
It is now clear that (v) and (vi) hold.
Hessian matrix is nonsingular. space is
(1
in a puncturec neighborhood of
Call the tangency of
(F,B)
over
H
XH
COl
reduced and irreducible, simple.
these, the first holds because
P, the
Since the two cones have
dim(X') - dimeV')
-
at
quadratic cone in
The second inequality follows from next two:
dim(X H ) + d
N-I
ill
denote th
to both the hypersurfaces
If
+ IpV
inequality in (iii) hold.
IICP,H)
s
, fr that define fl' N in a neighborhood of lP
valid and (ii) follows trivially. Since the sum,
0
is nondege
The form Let
CV , contradicting the hypothesis.
intersection, then
form
X 0 V , the Zariski tangent space is
TpX + TpV , were not al
Th
5.3, the tangent spaces
dim(X) + dim(V) - N
TpX 0 TpV , and the tangent cone is of codimension
an infinite linear family of hyperplanes
O.
works virtually without c
V)
(1
meet in
general theory of the sec
N-I
O.
V)
(1
CV
The result is a simp
is proper.
(P,H)
eN-I)
dim(X
and
V • H
(1
dim(X') + dim(V')
and
dim(X) + dim(V)
dim(X) + dimev)
If
CV sm .
(1
is isolated in
N-I
dim(TpX fl If
CXsm
V,sm) , then the tangency at
(1
dim(X) + dim(V)
(iii)
and
is isolated in
TpX + TpV ,and
(i)
(v)
(P,H)
0, whence
(P,H)
Crespo
P
A basic theorem abou
after Bertini.
It
says,
position, then they are t:
2 , and if the latter equivalent conditions are (3)
satisfied, then so is the third. The next result gives a useful criterion for a tangency to be simple.
BERTINI 1 S THEOREM.
group consisting of trans and in fact, disjoint if
(2)
THEOREM (Goldstein, pvt. comm., 840712). Suppose that (X sm 0 eV sm , and that H
lated point of the i:ltersect ion, Then the tangency at
(P,H)
is simple iff, at
(P,H) (3
is an iso-
X t sm 0 V' sm
(P,R) , the tangent spaces of
Indeed, it is a well, transitively on a variety dimension, then a general
AND II CV sm .
n
H
Then:
ex
and
VH •
at
(P,R)
>
N-
1
varieties in a third.
5.3, the tangent spaces of N
Zariski tangent space is
form Let
rand
(p ,G)
ilarly,
P must be
ev.
Thus (i) is
of
f 1-g ]
[f)
:H
-
I)
i
and because the XR X' , resp.
a point of
V is of dimension of
P ,because
(P,H)
(vi) hold. if it is proper, if ,ondegenerate double ent cone at
P
is
quadratic form whose N-l , then the tangent trivially satisfied. is satisfied, then X.V)
= 2.
Conversely,
lent conditions are gency to be simple.
n
0
f1
,gs
g J'
such that and
H
that define
V
as subschemes
is the hyperplane tangent at
=0
gl
Choose funct ions P
Then, in a Taylor expansion
II(p,H) .
II(p,H)
is nondegenerate, then it therefore defines a nondegenerate
a
containing the tangent cone of
X 0 V at
P.
simple.
(P,H) is an isoe X,sm fl V,sm
the tangent spaces of
So, by (1)(ii), the tangency is
Conversely, if the tangent cone is defined in
IpX 0 TpV
Q.
quadratic form, then clearly this form has to be induced by tangency is simple, then
IIep,H)
by a Hence, if the
is nondegenerate.
The preceding proof is of the same spirit as Ein's proof of 1-(25) and Katz's proof of 1-(17), although the work is observe how 1-(17) is a consequence of (2).
X.
hyperplane
H of
Suppose
is general.
H
hypersurface.
If
Obviously, Then
CH
!!lee t in
point
(resp.
is equal, as a scheme, to
is isolated in
iff at
0, whence iff t he scheme
(P,R) A
P
p),
(P,H)
ex
V to be a tangent
Namely, take
CX Q CH
(P,H)
It is good to
is isolated, then, by (2),
(P,H)
nondegenerate double point at
(l
CH
whence, by 1-(16), iff
CX 0 CH X
a
iff
X'
the tangent spaces of (resp.
is a
H has a CX
and
XH ) is the reduced
X' is ordinary.
basic theorem about tangency is the following one, conventionally named
after Bertini.
It says, in effect, that, if
X and
V are in general
position, then they are transversal, that is, not tangent. (3)
that
P
V
Since the two cones have the same dimension by (l)(iv) and since the former is
!cond holds because lS
By Goldstein's Prop.
iff the second fundamental
0
X and
reduced and irreducible, the two are equal.
dim(V') VH '
meet In
P, the constant and linear terms vanish, and the quadratic term
quadratic cone in
from next two:
CV TpX 0
X and
that define
at
Q defines
and
denote the codimensions of
in a neighborhood of
If and the first IS
s
to both the hypersurfaces
is any point in that and
lovely
may be obtained as follows; see Goldstein's Lem. 5.4
,H)
, fr 11' N
would lie in
ex
is nondegenerate on
lIep,H) The form
then it would lie in
?air
y 5)
The theory is developed over the complex numbers, but
works virtually without change over any ground field.
dim(X) + dim(V)
H
O.
general theory of the second fundamental form of the intersection of two smooth
O.
v)
meet in
199
The result is a simple application of Goldstein's ([1984],
is proper.
+ dim(V')
CV
DUALITY
BERTINI'S THEOREM.
group consist
There exists a nonempty open subset of the linear
of transformations
and in fact, disjoint if Indeed, it is a
g
such that
gX
and
V
are transversal,
dim(X) + dim(V) < N . fact that, whenever an algebraic group acts
transitively on a variety, given two subvarieties of less than complementary dimension, then a general translate of the first is disjoint from the second.
STEVEN L. KLEIMAN
200
F N , the second assertion
So, since the linear group acts transitively on holds.
group acts transitively as well on the graph incidence correspondence, (b) is of dimension gCX
where
By the same token, the first assertion holds because: (a) the linear
= egx.
CX
and
CV
I
of the point-hyperplane
N-]
are each of dimension
2N-1 , and (c) for any linear transformation
so, if and
I
is prime to
g, obviously
Thm. I, p. 153, and of that in Kleiman [1974], (10).)
C
S
]
then
,
whenever
V passes throu
The intersection mul
V varies in an
if
>
p
irreducible I-parameter family; the family need not be flat, nor the parameter space complete.
to be hoped that
Then the conormal varieties sweep out a variety
uev, the
(P,H)
Moreover,
CgX
these
intersects
if the family is nonconstant.
(P,R)
UCV
will lie in any given nonempty open subset of
0 , then the multipicity is a power
p
201
passes through the origin.
flat, nor the parameter UCV, the
is tangent at
The intersection multiplicity at if the characteristic
g
T
general I-parameter family of lines.
rre in Hodge-Pedoe [1952],
DUALITY
p , is such that every tangent line
I , then
=
AND
X : y = x S p+ I ,
(5)
THEOREM.
Let
V
vary in an irreducible I-parameter family. If g WN such that there are only finitely many
any linear transformation of tangent to
gX, then the number
#
of these
is V
V, each weight by a certain
STEVEN L. KLEIMAN
202
natural multiplicity, is given by the formula,
it where
rOIO
is the ith rank of
r'1
the family.
The number
/I
is
X and
o
M.
For Ii
is the ith characteristic number of
i
0, ... , N- I con!
parametrized by
JP I , of a
if either
< n-1
dim(X) + dim(V)
o=
+
+
or
< n-I
dim(X) + dim(V)
Clearly, the (closed) unio: Moreover, there exists a nonempty open subset of the linear group consist transformations
g
such that the number
is finite and each tangency is
it
proper and appears with the same multiplicity tangency is simple; if the characteristic then
q
and
bpe
where
b
q
p
if
=a ,
I , then each
q
then
q
is the number of bitangencies and
X and almost all the
of
V are reflexive, then
b
if
e >
o;
p
if
MN- i - 2 and H con Now, obviously, any linear is in
CX 0 cg-1V
> 0 P
*
2
Therefore, (7) and (4) yie
(8) Now, the formula
results immediately, by linearity, from (4) and from the following lemma.
LEMMA (Fulton-Kle iman-MacPherson . 1983], Lennna p. 14).
N-] , let
Li
be an i-plane. [CX]
trivial bundle
JPNxlPN*
3.3{b), p. 64), if classes of
JPN
h
and
0,
Then, modulo rational equivalence on
1,
and
[CX]
N-]
of the
1
of the hyperplane
differently.
N-cycles on
0, .•. , N-l
1II-3.
I.
I.
Now, by definition, 11-(1),
interpretation of the
Of course,
may be written as a linear combination of the , as asserted.
However, Schubert obtained the dual basis of cycles
In fact, he gave two similar and rather interesting
The proof of (6) leads as follows to another geometric riCX),
V , and
rj
basic, conditions, which r combining coefficients
, wi[CX]
For a lovely up-to-date version of this, see Grayson [1979J.
COMPLEMENTS.
ranks, II-{6).
condition should be exprel
derivations; one went via a determination of the Ktinneth decomposition of the relative diagonal of I/JP N ,and the other went via that of the absolute diagonal of
is properly tan
Schubert I s concept ion of for
one given by Schubert [1879], pp. 50-54, 289-295, for plane curves and curves I
is evident, if the V
The characterization
The preceding derivation of the contact formula in (5) is essential the
on
It
X iff
the choice of
denote the pullbacks to
[CLi] , and the combining coefficients are the
and surfaces in 3-space.
general linear pencil tang
multiplicities aside, (8)
1]
form the dual basis of the classes of (N-I)-cycles by 11-
Therefore,
is eq
h'
ri(X)
(11)(1).
r i (X)
equal; indeed, in both ehc
form a basis of the classes of (7)
PROPOSITION (Fulton-K
the ith rank
Hence, by standard theory (Fulton [1984], Thm.
hi(h I )N-i-l
[CLi]
rN_l(X)[
JPN* , then the elements
w·1
Hence, the
+
I/JP N is a subbundle of rank
To prove (6), note that
For
i
roeX)[CLoJ + .•.
CgX 0 UC
V are tang
I
To prove (5), it remains only to derive the formula.
(6)
onto
finitely many
Fix a complete flag, in which
Mj
is a general
a dual basic series of
others if, in every accep'
equal to the sum of the m
central theoretical probl, the basic conditions. For example, in the variety
V
in a
expressed as a linear cern spaces LO ' ... , 1; be found by impos ing the linear pencils.
Notice t
understanding of Schubert it, conditions would be i
AND
DUALITY
203
j-plane:
characteristic number of
f
f
>
Ti"
2 •
equality can be seen indi
d'" I
by 1-(15), and
q
Under specializati,
total number in both cases, but the two expressions of the number are equal,
(4).
term by term, but in reverse order.
might acquire an extranei
The self-duality of the formula in (5) is what led ehasles to discover it
X and the
in the case that
V are plane curves.
(186 J) had given a simple formula for the number
condition; namely,
=
If
rl , where, in effect,
Earlier, de Jonquieres
hypersurface of all curves of the same degree as
of
If
r
V satisfying a given
by the Contact Formula i: Formula. d.
In each formul
Since the two sets When
is the degree of the
p
V that satisfy the
1-(
15).
Thus de Jonquieres approach was in essence similar to that of Bischoff and
d
by
Steiner, which was discussed in the middle of
ported at
condition, and
is the degree of the curve parametrizing the given
1
§
V.
11-3.
Chasles felt, on philosophical grounds, that de Jonquiires's formula could not be right, because it was not self-dual. formula,
II
=
rOlO+rlll ,where
the condition, and family.
10
and
rO
and
rl
are numbers depending only on
He suggested that this formula applied without restriction on the
V are conics, and, although he did not offer a
condition when the
mathematical proof, he supported his contention with over 200 examples. higher plane curves
V
x.
The Bischoff-Steiner Formula, 11-(17), gives the class E
the f-fold Veronese embedding of
of degree It
e
rO
of the curve
by reembedding
of degree
f
parametrized by a general line
projective space of all curves of degree number of smooth
F
tangent to
E via
will now be shown that the formula
is a special case of the Contact Formula of (5): the case in which the F
q"
Consider a tang So
1-(10).
P.
scheme of length
F
a
It is
nc
d-l d-i
Since
q
and
dare b(
Bischoff-Steiner Formui
1
and is transveral. . finitely many
If
DUALITY
As to
?ally for the condition
Bischoff-Steiner Formula for not
E
rO(X)
X, or
E, to be smooth; the
was never explicitly used.
So, whether or
is smooth, the Contact Formula yields
lspired many geometers.
Lass
rO
of the curve
by reembedding
E
via
shown that the formula
,e in which the
leral line
L
V are in the
rulas count the weighted
: the weights might be
where F
and
are the characteristic numbers of the family of
is linear, simple direct considerations show that
conclusion above in the case that the Bischoff-Steiner formula; so
E
is a line,
I] = 2f-2.
I]
10
F
By the
may be calculated using
Thus the Contact Formula yields
the generalization of the Bischoff-Steiner Formula, II-(17a). III-3.
m-PARAMETER
Let
Since
V now vary in an irreducible but not
STEVEN L. KLEIMAN
206
necessarily flat m-parameter family, and consider the problem of finding the
II
(weighted) number number
of
V tangent to
XI' ••• , Xm .
varieties,
TIl
may be expressed in terms of the corresponding modules.
Ji,r
assertions may be proved
l
+
repeated a certain number T
Now, the m-
irreducible m-parameter family, and let gX 1, •••
XI' ... ,
Let
is finite, and if each
#
V
V vary in an
be given.
m linear transformations such that the number
multiplicity of appearance, then the
If not, then all
"inflated" characteristic
THEOREM (Fulton-Kleiman-MacPherson [1983], Thm. p. 6).
to
into the Hilbert scheu
image.
parameter version of (5) is this:
are any
If of
#
gl' ... ,
V tangent
is weighted by its natural
is given by the product of the modules of
Xi '
that degree. The following coroll indicates the content of
(I I) COROLLARY (Fulton-Kl at least one and at most tangent to
rCx l ) dX 2 ) •..
it
jo
11
jI
2 , a s
The corollary is iD1ll
... 1N-l
fact, exactly I)
by the corresponding characterisitic number of the family; that is, by the V tangent to is
(m;f)_1
intersection of degree> 10
it
=
curve of degree
jo' ... , jN-I ' by
replacing the monomial,
The number
m
provided that, for each
the product is evaluated by expanding it and, for all
number of
rxm.
parameter values, not the
r N_ I (X) IN- I
+
rOIO + rIll' associated to an arbitrary condition on conics.
gm
the action on
There is one wrinkle
The name "module" was given by Chasles (1864) to the formal expression,
(9)
The first part of (9) no.
1,
formal linear combination of indeterminates ro(X) 10
The
X is defined as the following
of (the condition to be tangent to) a variety
reX)
The
II
( 10)
ji
o
general i-planes for if, for some
< N-2
dim(X i ) + dim(V)
or
i
i
In the 19th century,
= 0, .. , , N-I .
was viewed as the primar)
V, either
and almost all
for m-parameter families,
< N-2
dim(X i ') + dim(V')
needed.
Finally, there exists a nonempty open subset of the m-fold self-product of the linear group consisting of m-tuples
(gl' ... ,gm)
such that
II
V pasE
A crude form of
the number of curves of ,
for the number of coni
is
finite and such that each tangency is proper and appears with the same
version of the procedure
multiplicity
one after the other, as
if
q
q
=0
1 , then each tangency is simple; if the
=
>
0 ,then
the number of distinct bitangencies and where
a ;
characteristic all
V and
p
Xl' •••
then ,
q
I ; if
P
e
are reflexive, then
if
Let
Pi
2
denote the projection of
is
and almost
S
birationally UCV Xffi
denote the
such that
points and lines.
(Pi,H i )
Cremol
care in this way; he rOIO + rIll' and
involving some negative
1
curves in terms of their the number of conics sub obscure conditions, to
of the set of (m+I)-tuples , (Pm,Hm) , V)
b
form,
equivalent to the parameter space of the family, and let
( (PI,H),
p
where
The most significant change
Choose any complete variety
closure, in the product
= bpe
b '" I
The proof of (9) is a lot like that of (5). is in the initial setup.
q
I
€
onto the ith factor.
CV sm Then,
surpr ise, found by Halph. The same procedure that in (5) as follows.
1
AND
?roblem of finding the XI' •.. , Xm modules.
The
•
DUALITY
1/
( I a)
The first part of (9) now follows immediately from (6), and the remaining
The module
efined as the following
assertions may be proved essentially as before, but using the transitivity of the action on
IXffi.
There is one wrinkle.
il
Strictly speaking,
parameter values, not the weighted number of
is the weighted number of
V
V might be
Almost all
repeated a certain number of times, namely, the degree of the rational map from
nmal expression,
:onics.
207
T
NOw, the m-
into the Hilbert scheme of
image.
FN.
Of course,
T
could be replaced by its
If not, then all the global intersection numbers, including the
"inflated" characteristic numbers, ought to be divided by their common factor, Let
. 6).
be given. II
JJIlber
V
vary in an
If
gl' ... ,
of
V
tangent
lted by its natural
reduct of the modules of
that degree. The following corollary is not at all obvious directly, and so it indicates the content of (9). (I I) COROLLARY (Fulton-Kleiman-MacPherson [1983], Cor., p. 8).
at least one and at most a finite number of hypersurfaces tangent to
m=
provided that, for each curve of degree
jo' ... , iN-I' by
subvarieties
Xl' •.. ,
i,
0
rO(X i )
V
There is always of degree
f
in general position,
(for example,
Xi
may be a point, a
2 , a smooth surface of degree> 2 , a smooth complete
intersection of degree
2 , or a smooth hypersurface of degree> 2 ).
The corollary is immediate, because obviously there is at least I (in fact, exactly I)
lly; that is, by the
, 0, ••• , N- I . all
It
[m(V')
for m-parameter families, (9).
< N-2
needed.
le m-fold self-product of II
is
P
the number of curves of a given degree satisfying several conditions, such as
b
is
and almost
points and lines.
lost significant change ,t
birationally UCV xm denote the
Cremona [1862], III bis. a, pp. 169-173, proceeded with due
care in this way; he began by rewriting form,
S
A more refined
one after the other, as replacements for the elementary conditions defined by
where 2
The latter was derived from the former, as
A crude form of this procedure was used in the first determination of
version of the procedure called for the introduction of the several conditions
lplej if the if
m general points.
6 5 for the number of conics tangent to 5 others; see § 11-3.
's with the same
q = bpe
passing through
was viewed as the primary result rather than as a special case of the formula
V, either
such that
V
In the 19th century, the Contact Formula for i-parameter families, (5),
rOla
+
the Bischoff-Steiner Formula in the
rll I ' and eventually arrived at a correct expression (Thm. XV,
involving some negative terms!) for the number of conics tangent to 5 smooth curves in terms of their degree. the number of conics
Similarly, Chasles arrived at a formula for
ect to 5 practically arbitrary conditions.
(Some
obscure conditions, to which the formula did not apply, were, to everyone's
Ipies
surprise, found by Halphen (1876).) The same procedure may be used to derive formally the formula in (9) from factor.
Then,
that in (5) as follows.
Consider the I-parameter subfamily of those
V
STEVEN L. KLEIMAN
208
tangent to
By (S), the number of these
X2 , •.• , Xm .
ro(XI)lo where
+
is the number of these
+
V
tangent to
V
x,I
is
S
rN-1lN-l
So, if the
[Z:
results applied to thei'
tangent to
L·1
translates
X2 , ..• , is, on the one hand, and on the other, by induction, given by the product, r(X 2 ) ... .
1·1
numbers will be I when I
The
O.
The number of these
V
Fix
[ZX i ] , is
admits an action of
orbits.
tangent to a general i-plane
i , and consider the (m-I) parameter-subfamily of those
particular
V tangent to
formula in (9) follows immediately.
For this derivation to be rigorous, suita-
ble care must be taken in the definition and treatment of the subfamilies. The factored form of the Contact Formula was not found
off; neither
[ZgX i ]
wit:
On the other hand,
total number of contact! position, and it yields III-S.
Cons ide
CONICS.
was that of Chasles's similar formula for the number of conics subject to S
suppose that the charact
conditions.
I, 2, 4, 4, 2,
The factorization of Chasles's formula was published by Prouhet
(1866) first, and then (independently) by Halphen (1873). inspired Schubert tremendously.
Halphen's note
Inde
So it suffices to verify
Schubert saw in the factorization more than a
conics.
Given a point
menemonic device; he saw in it a symbolic product of conditions, analogous to
conics through
what was done in symbolic logic.
hypersurface of
From Schubert's point of view, Chasles's
S-Condition Formula and the m-parameter Contact Formula are immediate
of
Giv
P.
of c
S
reembedded via th
H
consequences of the I-parameter formula; there is no longer any need for the
Steiner theorem, 11-(17)
cumbersome procedure of successive introduction of the several conditions.
2.
The traditional interpretation of Schubert's approach was advanced in the first half of this century by Severi and van der Waerden especially.
According
The linear group of
So ; the locus of line-p reembedded by the 2-fold
to this interpretation, the discussion of the Contact Formula sounds as
S2 ' (the Veronese surfa
follows.
contains only
Choose a convenient complete variety
the parameter space of the family. divisorial cycle on ( 12)
For each
S X
birationally equivalent to in
form the following
X.
Since no line can contail
[ZX]
and
P2* ( p]*[CX] . [UCVxI] )
So the number of
V tangent to
Xl ' ... ,
V
is equal to
provided this intersection number is defined. Formula. [CX]
[ZX]
like that for
[CX]
Of course, such an expression for
If it is defined, then an
in (6) will yield the Contact [ZX]
does follow from the one for
by the linearity of the right side of (J2), provided
[ZX]
is not
carelessly replaced by its reduction. To make the preceding derivation rigorous and complete, it must be shown that the
[ZX i ]
are locally principal (or at least the Poincare duals of
operators) and that they intersect properly; furthermore, the intersection multiplicities must be investigated. has been successful.
If
S
WP 3
Nevertheless, in practice this approach
is taken smooth, then any divisorial cycle, in
is disjoint fr,
divisor on
S 1 ' a 4-fol,
intersection of any 5 tr,
n
WP I
S [ZXIJ. .. [ZX N ] expression for
PI '
S
Its underlying set is the closure of the set of points representing the tangent to
S2
Fix S points
lies in
WP
n WP S '
So
By
t
ransv,
case, every point of into q
=
if
P
=0
and
q'
the weighted total numbe: p
#- 2
therefore
q
of 1, 2 and 4 distinct It is tempting now
I
verified, and in a certa
appl ied, then the characi verifying them is still
I
TANGENCY
.e
V
tangent to
is
Xl
particular orbits.
'e
V
Fix
tangent to
on the one hand, r(X 2 ) ...
DUALITY
[zxiJ , is locally principal.
209
Moreover, in some important cases,
admits an action of the linear group such that there are only finitely many
8
leral i-plane
AND
contain no orbit, then by the transversality
80, if the
results applied to their traces on each orbit, the intersection of translates
[ZgXiJ
m general
will lie in the open orbit, and the local intersection
Ii'
numbers will be I when the characteristic
p: 0
The
O.
always yields the finiteness of the
on to be rigorous, suitaof the subfamilies.
On the other hand, working on
total number of contacts, after the
Xi
and a power of
p
have been translated into general
position, and it yields information about the nature of each contact.
found right off; neither
111-5.
f conics SUbject to 5
suppose that the characteristic
s published by Prouhet
], 2, 4, 4, 2,
3).
So it suffices to verify the first three numbers. Let S be the W5 conics. Given a point P of W2 , let WP denote the hyperplane of
Halphen's note
actorization more than a onditions, analogous to of view, Chasles's
CONICS.
Consider the case of the family of all conics p # 2.
conics through
P. S
Given a line
H of
, let
WH
Steiner theorem, 11-( 17), and by I-(20)(i) and II-(2)(iv), The linear group of
has 3 orbits on
of
is the dual variety
W2
of
reembedded via the 2-fold Veronese embedding of
of S
denote the
H; that is,
of conics tangent to
onger any need for the
2 •
First,
Indeed, the dual family is also the family of all conics.
hypersurface of H
V
Then the characteristic numbers are
a are immediate several conditions.
p >
when
By the BischoffWE
is of degree
S : the locus of smooth conics, 2 W
)ach was advanced in the
So ; the locus of line-pairs,
"n especially.
reembedded by the 2-fold Veronese embedding; and the locus of double lines,
According
'ormula sounds as
52 ' (the Veronese surface).
itionally equivalent to
contains only
N
, form the following
Fix 5 points
PI' ...
Sinc e no line can cont ain
I ) representing the •..
V
is equal to
yield the Contact follow from the one for ded
[ZX]
is not
1ete, it must be shown Poincare duals of e, the intersection practice this approach divisorial cycle, in
Obviously
,
contains no orbit, and
and 2 lines
, Ps p]
WP
P2
and
HI' H2
WE
in general position.
, the intersect ion of
WP I
WP 2
and WP 3 is disjoint from S2' The trace of each WP i (resp. WEi ) is a divisor on S] , a 4-fo1d; hence, by dimensional transversa1ity on SI ' the intersection of any 5 traces is empty. WP 1
de fined, then an
S] , (whose closure is the dual variety of
lies in
So
n ... \)
Therefore, each of the 3 intersections
WP4 \l WE 1 '
By transversality on
SO'
each intersection is finite; in each
case, every point of intersection appears with the same multiplicity q
=
if
p: 0
and
q
=
pe
for some
e
if
p > 0
the weighted total numbers of points of intersection are p # 2 , therefore
q
=
of I, 2 and 4 distinct
I .
q, and
By Bezout's theorem, 1,2,4
Since
Thus the intersections are tranversal and consist
points.
It is tempting now to assert that the characteristic numbers have been verified, and in a certain sense they have been.
However, if (9) is to be
applied, then the characteristic numbers in the sense of (9) must be used, and verifying them is still tricky.
First note, that in both senses, the
STEVEN L.
210
The relevant char a!
characteristic numbers are numbers of smooth conics, and it is evident that a smooth conic appears in one count iff it does in the other.
no smooth conic is tanger
Hence, the
conic's tangent lines all
characteristic numbers in the sense of (9) are at least what they should be. If
o,
p
complete.
then the multiplicities are equal to
,and the verification is
Now, consider the intersection number (10). UCV XID
its value remains constant; however, acquire an extraneous component. p
e
Pi*[CXiJ
2 ; see below.)
from a general point, ane
may
p
( 10 + 211 )
> 0 than they are when
o
p
p
p *" 2 ) by applying (9) to the family of all smooth The number is finite, and since both ranks S = F5
of a smooth conic are 2 (for any
p) by 11-(10), the number is equal to
25
+ 2(1) +
+
numbers in both senses el
differ; moreover, the nUl characteristic.
+ 2(2) + 1)
p # 2,
The 3264 conics are distinct and the tangencies are simple if
3, 17 (these are the primes dividing 3264).
In fact, this is true if
p
= 3,
17 too, as is shown by a direct analysis made by Fulton-MacPherson; see Fulton
(A generalization of this
The above verification of the number 3264 is somewhat untraditional.
(although the local inte by (2),
It F5
consisting of a conic
V' . From this point of view, the absence of bitangencies and of
more points in common with a given conic form a proper closed subset of the set of all conics tangent to the given one. tangent to 5 general conics number higher order contacts.
e
2
CH 1 a
By II-( 10), both r. (21
V and
higher order contacts is obvious; there are none because the conics having 3 or
p
and
P4 *CH 1 and pS *CH 2 eac UCV x5 ); hence, the numb
along the Veronese surface -- the so-called variety of "complete-conics", (V,V')
CV
because V is not refle UCV x5 are not transvers
characteristic number mu
is more traditional (see Kleiman [1980]) to work entirely on the blowup of because its points represent the pairs
) 1 .
similar but more refined
to general
curves of higher degree is found in Hefez-Sacchiero [1983J, Cor., p. 8.)
Suppose the characteristic
V enume
The conic
with multiplicity
Each of the 3264 conics is smooth, and none is bitangent to one of the 5 given
its dual
Hence,'
as claimed.
3264
[1984], Ex. 9.1.9, p. 158.
The
Indeed, the analysis for
conics, parametrized by
conics.
according to Vainsencher little differently.
The number of conics tangent to 5 smooth conics in general position may
(21 0 +21 1 )5
5
From the point of v:
2 , and the
verification is compl.ete.
( 13)
is now a hJ
the line
In the case at hand, by what was
proved just above, the numbers are the same for all
now be computed (for
p *" 2 ; the only differet
Hence, in general, the characteristic
numbers in the sense of (9) may be less when (This does occur when
Under specialization,
or one of the
0
numbers are indeed
Conceivably, however, they are too large because the multiplicities are
0
+21
1
)5
An argument with tangent
ties of each of the enUD
conics, and each appear! 1980), via a more
Then the distinct smooth conics
51 , and there are no bitangencies nor
This fact was first proved by Vainsencher [1978], who
used an appropriate version of the variety of complete conics, namely the blowup of F 5 along the hyperplane (!) of double Alternatively, the
p
2 , half of the 3261
smooth conics in genera: 51 groups of 25 = 32
tangent to the reductim
matter can be treated as a residual intersection problem; see Fulton [1984J,
II 1-6 .
Ex. 9. 19, p. 158.
conics was generalized
OTHER CHARACTER
STEVEN L. KLEIMAN The relevant characteristic numbers now are
ld it is evident that a
:her.
Indeed,
conic's tangent lines all pass through a certain point; hence, the last three numbers are indeed
>
Iltiplicities are
md the verification is Under specialization, p.*[Cx.] J. J.
I, I, I, 0, 0, O.
no smooth conic is tangent to 3 or more nonconcurrent lines, because the
Hence, the
: what they should be.
the
211
0
The first three may be determined as above for
p # 2 ; the only difference is that the hypersurface the line
H·J.
is now a hyperplane. Only one line can be drawn tangent to a conic
Ie characteristic
I + 2(1) + p
of conics tangent to
from a general point, and (13) may be replaced by
may
.n they are when
WH i
0
=
at hand, by what was
# 2 , and the
according to Vainsencher [1978], p. 112. From the point of view of (9), the computation of the number 51 goes a little differently.
general position may mily of all smooth and since both ranks
umber is equal to
51
+ 0 + 0 + 0
The characteristic numbers now are
Indeed, the analysis for
p # 2
I, 2, 4, 0, 0, 0
made above still applies: the characteristic
numbers in both senses enumerate the same conics, but the multiplicities may differ; moreover, the numbers can only drop under reduction to positive characteristic.
Hence, the first characteristic number and the last three are
as claimed. The conic
V
with multiplicity to one of the 5 given
enumerated by the second characteristic number must appear
> I.
Indeed, the contact of
V
and
HI
is not simple
(although the local intersection number is equal to 2) because
p
=
2.
Hence,
by (2),
-MacPherson; see Fulton
CV and CHI are not transverse (this fact is also obvious directly, because V is not reflexive). Therefore, PI * CPl' ... , P4 * CP 4 , P5 * CHI' and UCV x5 are not transverse. Hence, the second characteristic number is 2 A
lalysis to general
similar but more refined argument shows that the conic enumerated by the third
33], Cor., p. 8.)
characteristic number must appear with multiplicity
p # 2,
s are simple if
his is true if
p
3,
=
lat untraditional.
It
lyon the blowup of
lP 5
By 11-( 10), both ranks of a smooth conic are 2. 2 5 ( I + 2(1) +
V and
)itangencies and of the conics having 3 or losed subset of the set inct smooth conics o bitangencies nor insencher [1978], who onics, namely the Alternatively, the see Fulton [1984J,
(the point is that
P4 *CHI and P5 *CH 2 each have a separate tangent vector in common with UCV x5 ); hence, the number is 4
'complete-conics" ,
19 of a conic
> 22
Hence, (13) becomes
25 .31
+ 0 + 0 + 0 )
An argument with tangent vectors, like those above, shows that the multiplicities of each of the enumerated conics is > 2 5 Therefore, there are 51 conics, and each appears with multiplicity 2 5 Higman (pvt. comm., spring 1980), via a more bare-handed procedure, also found that, under reduction to p
=
2 , half of the 3264 conics that are tangent, when
p
o ,
to 5 given
smooth conics in general position degenerate, and the other half coalesce, in 51 groups of 2 5 = 32 each, into the 51 distinct smooth conics that are tangent to the reductions of the 5 given conics. 111-6.
OTHER CHARACTERISTIC NUMBERS.
Chasles's enumerative theory of plane
conics was generalized right away to conics and quadrics in space especially by
STEVEN L. KLEIMAN
212
de Jonquieres (1864, 5), Chasles (1865, 6) and Zeuthen (1866).
Later Schubert
then finally those of the
(1894) began the development of an iterative procedure for finding the characteristic numbers of the family of all quadric f-folds in ]pN
the 12-parameter family
Schubert's work has now been fully and rigorously completed through the efforts
obtained the characterist
(1874, 5) refined this wo
of several mathematicians, who have studied the rich geometric structure of the
0
quartics via a lengthy at
variety of complete quadrics and the combinatorics of the algorithms for
sub fami! ies; a bout half
0
finding the characteristic numbers, etc. (and who have studied the closely
higher degree as well.
A
related variety of complete collineations, whose investigation was begun by
[1879].
Hirst (1873-1877) ).
Sterz [1982] introdu,
Van der Waerden (pvt. ms., 1981) worked out a relatively elementary theory of quadrics in
]p3, sufficient for the rigorous justification of another
famous number,
666,841,088, the number of quadrics tangent to
9
others.
Finat [1983] worked out an elementary local analytic treatment in higher dimension.
Vainsencher [1982J, [1984J gave a scheme-theoretic treatment, valid
in any characteristic, except 2 when in the case of quadrics; he proved that the variety of complete quadrics Crespo that of complete collineationsl may be obtained via a series of blowups starting from the variety of ordinary quadrics (resp. collineations) and he found the normal bundles of the centers.
Demazure
(pvt. comm., 810713) and De Concini-Procesi [1982], [1983J have studied "wonderful" equivariant compactifications of symmetric varieties of adjoint type in characteristic 0 (these are minimal among those such that the closure of every orbit is smooth).
De Concini, Gianni and Traverso [1983] implemented
the algorithm of De Concini and Procesi on a computer (in fact, on two for good measure); they obtained the characteristic numbers for quadrics in ]p4 and in F 5 and the number of quadrics tangent to 14 others in ]p4 and the number tangent to 20 others in]pS
No further cases
Laksov [1982] and (pvt. ms., fall 1982)
developed a scheme-theoretic version, set over an arbitrary base, of a
]p9
of all cubics via a s
coordinates.
Str6mmer (p'
enumerative results about
[1984], [1984a] verified I the nodal cubics; in both of embedded in lP 9 [1985] also have verified vindicating and advancing Schubert.
Ellingsrud (pVI
[1982], and Piene and Seh
compactification of the t1
to consider another appro.
an equivariant compactifil
soon be possible to say w Schubert found, the numbe surfaces.
Doubtless the
of the basic families of part of Hilbert's 15th
approach via linear algebra due to Semple (1952) and Tyrell (1956); Laksov recovered and advanced Vainschencher's results.
Laksov's work has been carried
one step further by Thorup-Kleiman [1985], who notably give the defining equations of the scheme of complete quadri.cs (resp. collineations). Chasles [1866], p. 326 top, wrote, 'Mais ce qui manque principalement, pour que la theorie des courbes d'ordre superieur soit aussi complete, ou du mains aussi avancee que celIe des con!ques, c'est de connaitre le nombres des
111-7.
13-14, derived it as fall of Conservation of Number varied. into
de toucher des droits ... 11 est probable que la connaissence de ces
of the
caracteristique, pour un ou deux ordres determines, mettrait sur la voie de la
curves
Maillard (1871) and Zeuthen (1872)
Degenerate
X i
tangent lines degenerates
courbes qui satisfont aux conditions elementaires de passer par des points et
10i generals pour un order quelconque ... "
SPECIALIZATION.
Contact Formula of (5):
rO
lines.
By defi
rO base points 0 V tangent to the
The preceding deriva
independently obtained the characteristic numbers of the 7-parameter family of
rigorously justified, as
all plane cubics, then those of the 8-parameter family of nodal cubics, and
specialization theorem.
TANGENCY 1
(1866).
Later Schubert
for finding the -folds:m
AND
213
DUALITY
then finally those of the 9-parameter family of all plane cubics.
Schubert
(1874, 5) refined this work and went on to find the characteristic numbers of
pN
the 12-parameter family of all twisted cubic space curves.
)leted through the efforts ieometric structure of the the algorithms for
Zeuthen (1873)
obtained the characteristic numbers of the 14-parameter family of all plane quartics via a lengthy step-by-step determination of the numbers of various subfamilies; about half of this work is theoretical in nature and valid in
studied the closely ;tigation was begun by
higher degree as well. [1879].
All this work is discussed to some extent in Schubert
No further cases are known.
Sterz [1982J introduced a variety of complete cubics, obtained from the _atively elementary theory ification of another angent to
9
others.
:reatment in higher he proved that the :ollineations) may be 'iety of ordinary quadrics Demazure
983] have studied varieties of adjoint
;e such that the closure Nerso [1983] implemented (in fact, on two for good quadrics in
n p4
p4
of all cubics via a series of 5 blowups, each described explicitly in Str¢mmer (pvt. comm., Feb. 1983) obtained some initial
enumerative results about cubics via some Chern class computations. Sacchiero [1984], [1984a] verified the characteristic
treatment, valid
of the centers.
p9
coordinates.
and in
and the number
. ms., fall 1982) trary base, of a lovely yrell (1956); Laksov v's work has been carried
of
p2
embedded in
p9
by the 3-fold Veronese map.
anque principalement, aussi complete, ou du
Kleiman and Speiser
1985] also have verified the characteristic numbers of the cuspidal cubics, by vindicating and advancing the original approach of Maillard, Zeuthen and Schubert.
Ellingsrud (pvt. comm., 1982), Harris (pvt.
COIDID.,
compactificat ion of the twisted cubics.
Abeasis (pvt. comm., 840607) has begun
to consider another approach to the enumeration of cubics, based on blowing up an equivariant compactification of the linear group of
surfaces.
Doubtless the rigorous determination of the characteristic numbers
of the basic families of varieties of degree
> 3 is the most important open
part of Hilbert's 15th problem. III-7.
SPECIALIZATION.
CO!:lsider, in the case of plane curves
V and
X, the
Schubert [1879], Beispiele 4, pp. 13-14, derived it as follows; he did so to illustrate the use of the Principle of Conservation of Number. varied.
Degenerate
X
By the principal,
into an rl-fold line.
remains constant when
ssence de ces
of the
rO
ttrait sur 1a voie de la
curves
V tangent to the reduction of the rl-fold line.
lines.
By definition of
pencils; in other words, 10 , there are
base points of the pencils.
X
is
Correspondingly, its set of
into
rO
rO
II
asser par des points et
of nodal cubics, and
Perhaps, it will
soon be possible to say with certainty that 5,819,539,783,680 is, just as
tangent lines degenerates into
he 7-parameter family of
p3.
Schubert found, the number of twisted cubics tangent to 12 general quadric
onnai tre Ie nombres des
871) and Zeuthen (1872)
1982), Piene
[1982], and Piene and Schlessinger [1983J have studied the Hilbert space
give the defining llineat ions).
numbers of the cuspidal cubics and
the nodal cubics; in both cases, he works with cycles on the conormal variety
10
curves
By definition of
X'
degenerates V through each
11 ' there are
11
The formula follows.
The preceding derivation may be generalized to arbitrary dimension, and rigorously justified, as follows. specialization theorem.
The first step is to prove the following
STEVEN L. KLEIMAN
214
(14)
[1984J, (2.5), p. 16; Henry-Merle-Sabbah [1984J, Cor. 4.2. I, p. 241). that the characteristic correspondingly,
CX
p
= O.
If
specializes to
X
is specialized to
Suppose
Xo ' and
CO' then, for suitable integers
2
[CoJ
row,
In other words, (14) says simply that every reduced, irreduci"::Jle cycle
Co
is of the form
CW
for a suitable subvariety
W of
XO'
In this
form, (14) may be nicely proved using the characterizat:on of conormal varieties as the Lagrangians (see § 1-2).
Because of the characterization, it
suffices to show that, at least when
p = 0 , if a variety satisfying a (first
order) differential equation is degenerated, then every reduced, irreducible component of the degeneration satisfies the equation too.
This lemma is
This proof works without change to yield (14) in the more general
case in which the ambient space variety
Y ;
is replaced by an arbitrary smooth
in fact, it yields the even more general case in which
varies smoothly.
The result is recalled by Sabbah [1983],
(3.2), and described by him there simply as a consequence of an (unspecified) Sabbah [1983], (4.3,2), goes on to prove an important
topological characterization (in terms of Milnor fibers and Euler obstructions) of the
Wand
row.
It would be good to have some sort of algebraeo-geometric
characterization of therrl. (3.9) ): the
A bare minimum is easy to prove (see Kleiman [1984],
W must include every component of Xo ; if a
component, then it must lie in the singular locus is a component that is not multiple, then
0
row
W is not a
f the scheme
if a
singular points
Xo
So, for example, if
X is
plus the set of hyperplanes through certain
Q, called
S
Fs
SW
in the 19th century; if
is reflexive iff Xs
degenerates tl
Co = CFO in JP2*
so Co is SI Indeed, othe
JP2* , but the image is The degenerat ion SI
called a homolography (I
case, is worked out in ( fix a complementary parameter family of all
CD
Ponto
H.
and base in
0 In
H, the
coordinate alone by a s, Under the homologr weighted sum
2 W prX
Then, since
), it suffices to prove, =
ri[CprX]
in
mH ,
of the support
fore, by II - ( 12) ( 1) ,
Ie,
for
N-2
it suffices to prove
X
(CX)p' viewed
is the dual of a curve of degree
p = 0 , and fix an arbitrary point
characteristic of lines of
pN
through
P
as a
TpX
P
P
of
> 2
Suppose that the X.
View the set
pN-l , and the projectivized normal (or
as a subvariety of it. through
hyperplanes of
that, for some N W
X , then the fiber
pN* , is the dual variety of the tangent space
THEOREM (Le-Teissier; see Teissier [1983J, Thm. §l).
tangent) cone
by II-(5).
:losure( b*c"d * [CX]) .
:I
a simple point of
cial case of the theorem, in which (18)
is numerically
K)
P
Hefez and Sacchiero [1983], Prop. :, p. 5, proved a refined version of the spesince
i[
CX): if
as a linear subspace of
[CprX]
It may be
viewed as a generalization of the following fact (which holds by the very definition of
TprQX
this new theory is the next theorem.
0
Correspondingly, view the set of
as the dual projective space
p(N-I)*, and
the fiber (CXl p the set (of limits) of hyperplanes tangent to as a subvariety of it. Then there exists a finite family of subvarieties
Wi
of
x
at
P --
PNpX, which
includes every irreducible component, and there exists a similar family of subvarieties
1,.;. ' l
of
such that
moreover, the conormal variety of irreducible components
CWo l
w: '
is the dual variety of
CW i' ) runs
the exceptional divisor
W· •,
through precisely the set E
of the blowup of
CX
STEVEN L. KLEIMAN
:220
along
(20)
(CX)p' In (18), the ambient space
FN
may be replaced by an arbitrary smooth
Y , as the proof below shows.
variety
Then the set of lines through
replaced by the projectivized tangent space of
Y at
P.
P
Let
is
Consider the degeneration of
is covered by the degeneration of X
if
CX
(More generally,
(CX)p
into the normal cone
N
into P
is replaced by
X
NCCX)pCX
R
NpX.
It
according to Sabbah
may be replaced by a smooth subvariety
CR Q CX.)
Hence, each component of
N
is a Lagrangian by the Specialization Theorem, (14); whence, so is each
E
component of
F(N)
Since
E maps onto both
FNpX
and
consider its Nash modi fica projectivized cotangent bu may be described as the ze Porteous's formula yields equivalence, for the cycle form of a formula known as induction on the dimension In the case
Lagrangians (see 1-2). The following theorem was first proved by Bruce ([1981], (2.12), p. 59) by more bare-handed methods under more restrictive hypotheses; Bruce required that
> 2N+3
X be a general hypersurface of degree
and that
N < 7.
Hefez (pvt.
corom.) observed that (19) is a corollary of (18). (Hefez-Bruce).
hypersurface of degree variety
X' .
Suppose that
> 2.
Let
NHX'
To prove (19), apply (18) with
X'
is a discrete set of points; indeed, obviously finite in view of 11-(8). Therefore, the The divisor X
at
to the
P
W·
Wi
H for
= CX
CX'
X and
by 1-(4), and
Wi'
Hence, the
are hyperplanes. E
X is a smooth
P.
Then
ex -> x'
(CX')H
most part so far (but cert
is
analytic, subanalytic, and
of (18) are points.
setting of algebraic geome
Thus (19) holds.
The jth "rank"
the weighted sum of the jth ranks
involving a wide and inter
The preceding discuss
would benefit the theory
of (18) is an interesting invariant of the singularity of ).
rj(W i )
the multiplicities of the "local polar" loci.
rj(W i
equal to MacPherson's Euler obstruction,
The
rj(E)
are
EuXCP) ,and
sum of the Sabbah [1983]
requires
p = 0 ; however,
information when
stratifications are The
>0
involves a Thorn-Whitney st
arb i. t rary, and it may yiel
These matters, their
relativization, and their connection with discussed in Henri-Merle-Sabbah [1984].
'».
p
0
For example, the
involves a version of Legr
rj(E) ,defined in (16), is then (resp.
it turns out in fact, i
the combining coefficients
gives only a taste of what
The multiplicities of its components are natural weights to assign W
As
Porteous's formula.
is a union of hyperplanes.
and
IPN ,
< N , then the cycles [CL
H be an arbitrary point of the dual
Then the normal cone
Y
diagonal, Lie algebra coho
o and that
p
Pu
and Giambelli (1905).)
(CX)p' the
assertion follows from the characterization of conorrnal varieties as the
(19)
generate, modulo rat The idea of the proof
To prove the general form of (18) is simple and easy as follows (Sabbah,
R of
be arb
W run through any se
[CW]
generalization is not as important.
[1983], (4.4, 2).
p
modulo rational equivalenc
Le and Teissier
considered this case; however, their work is local analytic, and then the
pvt. comm.).
THEOREM (Fulton-Klei
characteristic
rj(EJ
is
gives an
interpretation of MacPherson's calculus of constructible functions in terms of
p >0 .
connected with the
these 1ines will lead to a
Whitney stratifications wh
geometric operations, such as intersection, pushout, etc., on conormal cycles
interesting will come out
(these are the linear combinations of the fundamental cycles of conormal
and duality over arbitrary
varieties).
In this connection, the next result may be of interest.
TANGENCY (20)
of lines through t
P
Le and Teissier
P.
p
be arbitrary, and let
generate,
modulo rational equivalence, the group of all cycles classes on
Y.
Then the
W run through any set of subvarieties of
generate, modulo rational equivalence, the group of all conormal cycles. The idea of the proof of (20) is this.
easy as follows (Sabbah, the normal cone
K)pCX
NpX.
It
according to Sabbah
by a smooth subvariety :e, each component of
N
whence, so is each
FNpX
and
consider its Nash modification
( [ 198 1J, (2. 12), p. 59) by
N < 7.
Hefez (pvt.
and the pullback to
Y.
Y ,
X of
of the
In the latter, the closure of
may be described as the zero locus of a regular map of bundles. equivalence, for the cycle of the closure.
CXsm
Applying
(Porteous's formula is a modern
form of a formula known as the formula of Special Position to Schubert (1903)
In the case
:heses; Bruce required that
Given a subvariety
Porteous's formula yields a certain expression, valid modulo rational
Pushing the expression out over
Y = JPN , (20) implies that, i f
N , then the cycles
Y
and applying
X yields (20).
induction on the dimension of
nal varieties as the
:hat
X@
projectivized cotangent bundle of
and Giambelli (1905).)
(CX)p, the
Let the
Y be an arbitrary smooth variety.
[wJ
[CW]
aalytic, and then the
221
Y whose cycles
Let
is
DUALITY
THEOREH (Fulton-Kleiman-MacPherson [1983J, Prop. (c), p. 24).
characteristic
I by an arbitrary smooth
AND
generate.
Li is an i-plane for
0 < i
This is again the main content of (6).
As it turns out in fact, in the present case, the preceding proof also yields the combining coefficients.
Thus there is a richess of proofs of (6),
involving a wide and interest
range of methods: Kunneth decomposition of the
diagonal, Lie algebra cohomology, cellular decomposition, degeneration, and
1d that
X is a smooth
Porteous's formula.
point of the dual
1
The
of hyperplanes.
discussion of the new connections with singularity theory
gives only a taste of what has been done. X and
-(4), and
P.
Then
CX -) X'
(eX')H is
However, what has bee.n done, for the
most part so far (but certainly not entirely), has been in the setting of analytic, subanalytic, and semi-analytic geometry and only incidently in the
of (18) are points.
setting of algebraic geometry.
Doubtless a more algebraeo-geometric approach
would benefit the theory of singularities not only when of the singularity of natural weights to assign
'».
The
rj(E)
are
Sabbah [1983]
rj(E)
is
gives an
.ble functions in terms of etc., on conormal cycles cycles of conormal be
0
f interest.
However, the algebraeo-geometric proof For an arbitrary degeneration, (14)
p = 0 ; however, the case of degeneration to the normal bundle is not
information when
'y stratifications are
.d
but also when
arbitrary, and it may yield to a direct analysis and provide interesting
matters, their
:ing sum of the
requires
p = 0
For example, the analytic proof of the Specialization Theorem, (14),
involves a Thorn-Whitney stratification.
involves a version of Legrangian geometry.
refined in (16), is then j(W i
p ) 0
p > O.
The Le-Teissier Theorem, (18), is so intimately
connected with the Whitney conditions that it is probable that proceeding along these lines will lead to a new a:gebraic treatment of the theory of ThomWhitney stratifications when interesting will corr.e out when
p
o
and it is possible that something new and
p) 0
The future of the theory of tangency
and duality over arbitrary fields may well lie here!
222
STEVEN L. KLEIMAN
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Ann. Sc i. Ec.
DUALITY
2-278
MA 02i39 Current Address: Mathematics Institute Universitetsparken 5 2100 Copenhagen 0 , Denmark
