dimension at least g - 12 and is made up by base point free webs gg_~a of. Clifford index 9 - 8. If none of these is primitive the same reasoning shows then that ...
manuscripta math. 77, 237- 264 (1992)
manuscripta mathemati ca 9 Springer-Verlag 1992
PRIMITIVE
LINEAR
SERIES ON CURVES
Marc Coppens, Changho Keem and Gerriet Martens
In this paper we study a new numerical invariant ~ of curves C whidl is related to the primitive linear series on C. (Primitive series - defined below - are the essential complete and special linear series on C.) The curves with/~ _< 3 are classified, and it is shown that for a given value of s the curve is a double covering if its genus is sufficiently high. The main tool are dilnension theorems of It. Martens-Mumfordtype for the varieties of special divisors of C, and we prove two refinements of these theorems.
Introduction (0.1) Let C denote a smooth irreducible projective curve of genus g > 0 over C. The basic theorem for tile study of linear series on C is the theorem of Riemann and Roch. In the interpretation of Brill and M. Noether, as a reciprocity law, it says d - 2r = d' - 2r' if 9~ is a complete linear series of degree d and dimension r on C and g~; is its dual series life - g~a[. IIere, of course, IKcl denotes the canonical series of C. Since d + d' = 2g - 2, given d we can compute r iff we know r t Clearly, r' > - 1 , and if d > 0 the series g~ is called special. So, for the question which linear series exist on C for a fixed degree d we can restrict ourselves to special series of this degree (if any), and obviously it is enough to consider only those of them which are base point free and for which also the dual series are base point free. (In fact, otherwise the g~ comes from a unique g~a-i resp. a unique gd+j _r+j by adding i --> I rcsp. subtracting j > -- 1 appropriate points of C.)
237
C O P P E N S E T AL. A complete and base point free linear series g~ on C is classically called primitive if its dual series is base point free, too. The trivial primitive series goo and its dual [I(c] are always present, and so we disregard them in what follows. Specifically, we let g > 4 to insure that there are non-trivial primitive series on C. The expression d - 2r from above is called the Clifford index of the g~. For a primitive series it is a non-negative number which can be interpreted as a numerical measure to which extend this series is "unusual" for a curve of genus 9: The smaller this number, the more global sections the series has for its degree. It is natural to attach to C its Clifford sequence S = S(C) obtained by ordering the Clifford indices of all (nontrivial) primitive series of C, omitting repetitions. The first (i.e. the smallest) member of S is known as the Clifford index c of the curve C. By Brill-Noether-theory, if C is the general curve of genus g, it has the "ordinary" Clifford sequence S = (c,c + 1,c + 2 , . . . ,9 - 3) without gaps where c has its largest value it can have for a curve of genus 9, namely c = [~-~2a]. On the other extreme, if C is "most special" with respect to linear series, i.e. if C is hyperelliptic, we have S = (0). (Recall that this does not mean that a hyperelliptic curve has only one primitive series; it just says that all its primitive series have the same Clifford index 0.) We call the length g = g(c) of S the primitive length of C, and we consider it as an invariant of curves of fixed genus g and of fixed Clifford index c (or, what is nearly the same ([CM]),'of fixed gonality). (0.2) In this paper we want to study primitive linear series, and the entities S and g related to them. In the first part we gather preliminary results. Specifically, we detect all primitive series on trigonal curves, and we discuss when multiples ng~ (n E N) of a complete and base point free 9~ are primitive. To control the related Clifford indices we shall make use of the following principle (cf. [ELMS], Prop. 3.1): If D and E are divisors on C such that IEI is base point fi'ee and 12D - E[ is special then cliff(D - E) < cliff(D). In the second part we consider curves of small primitive length. In particular we will show:
(1)
For g > 7 we have g = 1 ~ff C is hyperelliptic .
(2)
For 9 > 11 we have e = 2 iff C is bi-elliptic (i.e. a double cover of an elliptic curve}.
(3)
For 9 > 19 there are no curves of primitive length g = 3.
(4)
If C is a double cover of a curve C' then g < c(g') where c(g') is a consta,',t depending only of the genus 9' of C'.
238
C O P P E N S E T AL. The main tool in proving these results are dimension-statements for the varieties lvV~ of special divisors of C, on the Jacobian of C. To generalize these results, we will prove in the third part a new dimension theorem for W~, i.e. for nets. This enables us to show (5) Let n E N, g >_ 2 ( n + 1 ) 2 and dim W~+ ' 3 < 1. Then C has primitive pencils of degree g - n , . . . , 9 - 1. (6) A curve of a given primitive length cannot have arbitrary high genus unless it is a double covering. We also prove another dimension theorem for W~ which generalizes both tI. Martens' well-known theorem and the Accola-Griflqths-IIarris-theorem on this subject. Our theorem characterizes double coverings (cf. Theorem (3.2.1) for details), and in a final digression we will use it to classify double covers by another interesting invariant of curves which is related to the singularities of the theta divisor. We have already noted that we view the primitive length g as an invariant of curves of fixed genus 9 (9 -> 4) and fixed gonality k. It would be very interesting to (dis-)prove that g is lower-semicontinuous on the irreducible sublocus ,A4g,k of isomorphy-classes of k-gonal curves in :V/g, the moduli space of (smooth) curves of genus g. In a forthcoming paper we will study primitive series on the general k-gonal curve of genus 9 (i.e. the general element of .Adg,k), for 3 < k < ~ + 1. (0.3) N o t a t i o n s a n d C o n v e n t i o n s : C always denotes a smooth, irreducible, projective curve of genus 9 -> 4 over the complex numbers C. The gonality of C is the minimal degree of morphisms from C to ~i; so in some sense it measures the "degree of irrationality" of C. We will use the following notations for divisors, linear series, invertible sheaves, line bundles etc. on C: As usual, a g~ is an r-dimensional linear series of degree d on C. (It need not be complete but in most cases in our paper it will be.) For r = 1 (r = 2, r = 3) it is called a pencil (net, web). If D is a divisor on C we write IDI for the associated complete linear series on C. By K c we denote a canonical divisor on C, and [lfcl is the canonical linear series of C. By a point P on C we always mean a closed point of C, i.e. an effective divisor of degree one on C. We write g~ + g~ (resp. tg~ , 19~ + DI, [D - g~l if t E N and D is a divisor on C) for the complete linear series [E + FI (resp. [tEl, [E + DI, [D - El) where E E gS, F E g~. Note that one has to distinguish between [g~ + DI and the subseries g~ + D. The series ]Kc - 9~1 is called the dual series to g~. Finally, for a linear series g,] and a divisor D on C we let g S ( - D ) be the linear series on C made up by all divisors E - D for E E g,] such that E >_ D (i.e. such that E - D is effective). Clearly, 9 5 ( - D ) = Ig~ - DI if g5 is complete.
239
COPPENS ET AL. If L is an invertible sheaf (or a line bundle) on C and i = 0, 1 we abbreviate Iti(C,L) (resp. dimHi(C,L)) by fli(L) (resp. h'(L)). In particular, for a divisor D we write H'(D), hi(D) instead of H'(C, Oc(D)),dimIl'(C, Oc(D)). Let J(C) '~ Pic~ be the Jacobian variety of C, identified with the Picard variety of C parametrizing line bundles of degree 0 on C. Clearly, J(C) is an abelian variety of dimension 9 and it will cause no confusion to denote the addition on J(C) by +. So for two non-empty subsets A and B of J(C) we set A 4- B := {a 4- bia E A, b E B}. Chosen a fixed reference point P0 on C we have the classical morphism I(d) which maps an effective divisor D of degree d of C onto the point of J(C) associated to Oc(D - dPo). Then W~ := {x E
J(C)ldimI(d)-~(x) >_v}
is a Zariski-closed subset of J(C) parametrizing tim complete linear series of degree d and dimension >_ r on C. In the case v = 0 we simply write Wa instead g-1 of W ~ For example, W~g_~ consists only of the point n = nc := I(29-2)(Kc) which is called the canonical point on J(C). If x E W~ \ W~+' we let g2(x) be the complete linear series I(d) -~ (x) of dimension v on C. A g2 on C is called base point free if dim(95(-P)) = r - 1 for any choice of P on C. It then defines a morphism f : C ~ pr onto a non-degenerate irreducible (but maybe singular) curve in Pr. If f is birational (resp. an isomorphism) onto its image curve f(C) the 92 is called a simple (resp. a very ample) series. (Some authors prefer the phrase birationally very ample instead of simple.) Let C' be the normalization of f(C), and assume that our g2 defining f is not simple. This means that g2(-P) has base points for any choice of P on C. So tile induced morphism f : C ~ C' is a non-trivial covering map of some degree k > 2, and if g~ is a linear series on C' then f*(9~) = {f*(D')ID' E g~} is a linear series on C of degree kn and of the same dimension s which is called induced by f (or by C'). For example, our g2 defining f gives us a 9~Ir on C' and is thus induced by C'. Finally, if g' is the genus of C' we have g' < g (by the Riemann-Hurwitz relation since we assume g >- 4), and the g2 is classically called compounded of an involution of order k and genus g'. In the case g' > 0 we speak of an irrational involution on C. For 0 < x E R we denote by [x] the integer part of x, i.e. the biggest integer ~-lweseethat --(2n - 1)D is a special divisor. From [ELMS], Lcmma 3.1 (cf. (0.2)) it follows that cliff((n - 1)D) < cliff(nD). For k _< 3 this inequality is a consequence of Example (1.1.2) resp. of the following Example (1.2.7). 9 Recall that a base point free g}, on C is called compounded if the induced rnorphism C --* ~,1 factors nontrivially through a morphism C --, C' onto a Curve C t. (1.2.4) C o r o l l a r y : Let g~
(k > 3) be a complete and base point free pencil on C. Assume that k is odd or that the 91k is not compounded. Then C has r2-L~2-] primitive length e >_ tk(k-1)J" Proof: Let g~, = 29-2 that n _< k(k-1)"
IDI.
We may assume that k(k - 1) < 29 - 2. Let n E N such
Let k > 3 be odd. Assume that there is a m < n such that cliff(roD) = cliff(nD). By Lemma (1.2.3), then, c l i f f ( ( n - 1)n) = cliff(nD), i.e.
(n - 1)k -
2h~
1)D) + 2 =
-
nk
-
2h~
+ 2,
and so k is even. Tiffs contradiction shows that all the series InD[ (n < k-1)J have different Clifford indices. Since they are primitive (cf. Lemma (1.2.3)) we have/? >- - rI . k (2g-2] k-l) J " Let k > 3 be arbitrary but assume that the g~ is not compounded. If we can show that we have then h~ series ng~
= n + l for n < ~k ( k - 1 ) all the primitive -
(k >_ 3) have different Clifford indices whence g > tk(k-1)J again.
So assume that there is a n -< ~k ( k - 1 ) such that ]nD[ has dimension r > n + 1. Then this series defines a morphism f of degree m onto a curve C' of degree "~ in E '~. By the Riemann-Roch theorem, applied to the induced series g ~ on m m
the normalization of C', we have
nk
n+l
< r
0 the map C ~ P1 of degree k given by the g~ factors through f . By assumption, the g~ is not COmpounded. Since m < k we thus must have m = 1, i.e. f is birational. But then we obtain, by Castelnuovo's genus formula, g_< ( k - 1 ) ( n k - l - l k n ) = l k ( k - 1 ) n - ( k - 1 ) , ~5
L
243
C O P P E N S E T AL. a contradiction to our choice of n.
[]
(1.2.5) C o r o l l a r y : Let k > 3 be an odd number. Then a k-gonal curve of a given primitive length cannot have arbitrary high genus. [] (1.2.6) R e m a r k : The last part of the proof of Corollary (1.2.4) shows that we have the bound el >_ t k(k-l) J -- 1 for the first scrollar invariant el of an uncompounded g~ on a k-gonal curve (k _> 3). Note that Corollary (1.2.4) applies to the general k-gonal curve of genus g but in this case the bound for g given in Corollary (1.2.2) is better. [] (1.2.7) E x a m p l e : Let C be a trigonal curve of genus g > 4. Then the 9~ on C is unique, and so its first scrollar invariant el is an invariant of the curve C which is called Maroni's invariant and usually denoted by m. We have a_vA < m < ~2 ' and rn completely determines the distribution of linear 3 --series on C ([MS]). Since rg] : g;r for 1 < r < m + 1 we obtain from Lemma (1.2.1) that the multiples rg~ of the g3~ are primitive for r = 1 , . . . ,m. We will show that these multiples of 913 and their dual series are the only primitive series on C, unless m is minimal, i.e. m = ~3 ' in which case also the series ( m + 1)g~ of degree g - 1 is primitive. To prove this we use the description of I,V~ (d < g - 1) given in [MS], Proposition 1: If we set W~ = { r } , U j := rT"+Wd_3, we have W~ = Uj U (t~- Uj:) for d - 2 r = d ' - 2 r ' and d + d ' = 2 g - 2 , and to-U~: (if non-empty, i.e. if d' > 3r') is an irreducible component of W~ different from U~ if and only if r' _< m. Thus, a complete and base point free g,] (d < g - 1), ]D[ say, different from rg~ belongs to - U~:, i.e. life - D[ belongs to U~;, and if the g~ is primitive we must have d' = 3r', i.e. [ K c - D [ = r'g~ (r' < m). For m > 9-4 tile series g~, 2 g ~ , . . . , mg~ cover all multiples of g~ of degree at most g - 1, and so only g~, .,mg~ and their duals are primitive. Let m be minimah m = ~ Then (m + 1)g~ has degree g - 1, and r' = r = m + 1 = ~3 is bigger than m. This implies ~ ; - U"9 - 1 = U9--1 ~ = {rr}, i.e. (m + 1)g~ coincides with its dual and is therefore primitive. So C has the Clifford sequence S = (1, 2 , . . . , m) resp. S = ( 1 , 2 , " ' " , m + l ) i f m > ZZAresp. m = ~:A [] 3 3 --
"*
3
3
~
The Maroni-invariant is lower semi-continuous in families of trigonal curves. Thus, by Examples (1.1.2) and (1.2.7), r is lower semi-continuous in families of 2 - g o n a l resp. 3 - g o n a l curves of genus g. More generally, we like to ask the (1.3.1) Q u e s t i o n : Let Mg,k be the (irreducible) moduli space of k-gonal curves of genus g. Is e lower semi-continuous oll Mg,k? (1.3.2) E x a m p l e : Let C be an extremal space curve of degree 8. Then g = 9, the unique and very ample 93 on C is half-canonical (i.e. 2gs3 = [I(v]) and C lies on a unique quadric Q. By [All, Lemma 2.4 we have dim(gs3 - 91) >- 3 -
244
COPPENSET
AL.
for any base point free g~ on C. Thus (cf. [ACGII], IV,ex.F-2): =
+ w,
W~ = (W8a - W2) O (W~ + W1) =
W~ = r
(note g] is very ample)
w} : w 2 - w,
+ w,
Claim 1: W~ consists of two points, p,q, say, and Ws~ = (W;? + W1) U {2p} t0 {2q} if Q is smooth. (C is then of bi-degree (4,4) on Q ~ ~1 X ]I1~1.) W4x consists of a unique point (nonrcduccd), and W~ = l/V;?+ W1 if Q is a cone. Proof: The points of W2 clearly correspond to the rulings of Q. So there are two different pencils of degree 4,g~ and h~, if Q is smooth, and there is only one g~ if Q is not smooth. We have g~ = g~ + h~ (with h I = g~ if[ Q is a cone). Assume there is a complete and base point free gs2 and g~ r 2g4,gs I # 2h 1 (these inequalities clearly hold in the cone case). By the base point free pencil trick we have (cf. [ACGH],III, cx. B-4): dim(g~ + gl) + dim(g8~ - g4a) -> 4. If d i m ( g 8 2 - 9 ~ ) = - l w e g e t g s 2 + 9 1 =g~2 =
( [IVc-g~' = g1+ 2h~ } ]I(e hZ4] = 2gZ4+ hZ4 since
[Iic[ = 2g~ = 2.q~ + 2hi. Thus we obtain the contradiction .q~ = 2h I resp.
g~ =
gl +
hl
= g~. If dim(g~ - g l ) >-- 1 we would have g~ =
gl + hl = g2
"
So we obtain dim(g~ - g l ) = O. Our g~ cannot be simple because otherwise [g~ - P[ would be a base point free g~ on 6', for a general P e G'. We have the commutative diagram C
C ! C ]~2
f
~. g~
projection
p1 where f is induced from g~. Note that 1 < deg f < 4. Thus deg f = 2, and C is a double covering of a curve of genus < 2. But this contradicts [All, lemma 2.6. Claim 2: If Q is smooth C has (up to duality) only the primitive series 1 I 1 I 3 g4,h4,2g4,2h4,gs and (X)5 pencils gs~. So the Clifford sequence is (2,4,6) and 2=3. if Q is a cone C has (up to duality) only the primitive series gl, 2gI = g~ and ~s pencils g8a. So C has the Clifford sequence (2, 6) and length s = 2. Proof: The primitivity of cx)s pencils g~-z is proved in Lcmma (2.1.1). For the primitivity of the other mentioned series of degree 8 we need only note that
245
COPPENS E T AL. w;
= r +
wg - w,
= ,, - (
+ w, ) =
- ( w2 -
+
) = w2 - wi +
=
Note that there are families of extremal space curves of degree 8 having two different pencils of degree 4 in general and specializing to such a curve with only one 94a ([Col]). We have seen that this specialization makes s drop by 1. There are also families of curves of bi-dcgree (4,4) specializing to a bi-elliptic curve ([M4], (4.2)). Again, we will see in the next w that g drops by 1. 9
2
C u r v e s of p r i m i t i v e l e n g t h < 3
The first member of the Clifford sequence of C clearly is the Clifford index c of C. Our next result shows that the last member usually is 9 - 3: (2.1.1) L e m m a : Let c _> 2. Then G has a primitive 9g-1.1 P r o o f : Assume there is no primitive g~-l. Then W~_ 1 is the union of W~_2+W1 x is of dimension at least g - 5. According to a n d s - ( W g1- 2 + W 1 ) whence Wg-2 the Martens-Mumford theorem ([ACGII], IV, w C is then of Clifford index c < 1 unless C is bi-elliptic. By our hypothesis, C is bi-elliptic, g = 4 would imply c = 1. For 9 = 5 any 941 on C is primitive, by Example (1.1.4). So let g > 6. Then W91- 2 = W~ + Wg-6, and it is known that the irreducible component W~_ 2 + W1 = W~ + Wg_5 of WJ_ 1 coincides with the dual set - ( W ,-2 1 + I'VI) and that W',_~ is not irreducible (e.g. [ACGtI], VIII, ex. F). Thus there is a primitive gg-1 l on the bi-elliptic curve C. [] (2.1.2) P r o p o s i t i o n : C is hyperelliplic.
Let g >- 7. Then C has primitive length 1 if and only if
P r o o f : By Example (1.1.2) c = 0 i m p l i e s s 1. If c = l a n d g _ > 7 Cis trigonal and g > 2, by Example (1.2.7). Let c >_ 2. According to Lemma (2.1.1) C has a primitive 9g-x, 1 and f o r g >_6wchavecliff(g~_x) = g - 3 > g-12 -> c ' So C has length g _> 2. [] (2.1.3) R e m a r k : A non-hyperelliptic curve C of genus 6 has g = 1 if and only if C is trigonal with Maroni's invariant m = 1 (i.e. without base point free g~) or a smooth plane quintic. In particular, the genus bound in Proposition (2.1.2) is best possible. [] (2.2) Before we turn to curves of primitive length 2 we state a technical result which has remarkable consequences for curves which are double covers. Recall (cf. (0.3)) the following notation: If f : C ~ C' is a non-trivial covering of smooth curves we say that a 9~ on C is induced by C' if gl = f'(g~) for a g~ rl
n
on C', where n = deg f _> 2. (2.2.1) L e m m a : Let f : C ~ C' be a non-trivial covering of smooth curves and g'd (r >__ 1) a base point free linear series on C which is not induced by
246
C O P P E N S E T AL. C'. Let P x , . . . , Pr-1 be r - 1 general points on C. Then the base point free part of the pencil gS(-P1 - . . . - Pr-1) on C is not induced by C'.
Proof: Clearly, we may assume r >_ 2. By induction, it is enough to show that for general P E C the base point free part of the (r - 1)-dimensional linear series g~d(--P) is not induced by C'. Consider the finite and non-trivial field extension of the function fields of C, C' given by f : C ( C ' ) ~ C ( C ) . Fixing a divisor D E g~ the linear series g5 corresponds to an (r + 1)-dimensional C-vectorspace V C H~ C C(C). The fact that g~ is not induced by C' is equivalent to the statement V ~ C(C'). Let PI,P2 E Supp(D) such that P2 is not a base point of g~d(--P1) and let r p.~)" V~ (i = 1,2) be the r- dimcnsional linear subspace of V belonging to gd(-Since 88 ~ V2 we have V~ q- V2 = V. So V1 ~ C(C') or V2 ~ C(C'), i.e. for at least one of the series g d ( - p,i) its base point free part is not induced by C I. 9 We apply this lemma to double coverings. (An analogous but more complicated result holds for coverings of prime degree.) (2.2.2) C o r o l l a r y : Let C be a double covering of a curve C' of genus g'. Then a base point fl'ee linear series g~ on C of degree d < g - 1 is induced by C ~ if d < g - 2g ~ or if r >_ 2g'. Proof: For d < g - 2 g ' the statement is well-known (e.g. [M1], w Ifd < g - 1 and r > 2g I choose r - 1 gcneral points P 1 , . . - , P~-I on C. Then the base point free part g~ of g~d(--Px - - . . . -- Pr_~) has degree e < d - r + 1 < g - 1 - 2g' + 1 = g - 2g ~ and so is induced by C'. By the lemma, then, our g~ is induced by C ~. The following result shows that the primitive length of a doublc covering C of a fixcd curve C ~ remains bounded if g ---* oo. This may be rephrased by saying that double coverings have bounded length. (2.2.3) C o r o l l a r y : Let C be a double covering of a curve C' of genus g' > 1. Then C has primitive length e 2g'. If d
3 , we have cliff(g
) __ d - 2 < 4g' - 4. Since cli r(g ) _> 0 is e v e n (recall
float g~ is induced) there are at most 2g I - 2 possibilities for the Clifford index. It d E S~ the inducing series g~ of C ~ must be nonspecial (being of degree 2
247
C O P P E N S E T AL. d ~91" _> 2 9 ' - 1) whence r = ~This implies constant Clifford index d - 2 r = 29' on $2. If d E 5'3 and r >_ 29' the inducing series of C' again must be nonspecial (since r >_ 9'), and so the Clifford index of 9~ is 29'. For the remaining case d E $3 and r < 2g' an easy computation shows that we have at most 6g' - 5 different Clifford indices. Thus g < (29' - 2) + 1 + (69' - 5) = 89' - 6. []
(2.3.1) P r o p o s i t i o n : Let 9 >- 11. Then C has primitive length 2 if and only if C is bi-elliptic (i.e. a double coverin 9 of an elliptic curve). P r o o f : According to Lemma (2.1.1) and Corollary (2.2.3) a bi-elliptic curve C of genus 9 >- 6 has the Clifford sequence (2,g-3) of length g = 2. (Note that a smooth plane quintic cannot be bi-etliptic, c.f. the proof of Lemma (2.1.1).) Conversely, we will prove that g = 2 implies that C is bi-elliptic unless g _< 10. Let 9 > 9. Then a refinement of the Martens-Mumford-theorem (on the dimensions of W~'s) says that dim W~_3 > g - 8 implies c < 2 ([K] for 9 _> 11, [Co2] and [Col], w for g = 9 and g = 10). Assume c > 3. Then dim WJ_, = g - 2 ( i + 1 ) for 0 < i < 3. If dim Wff_2 _> g - 8 we would have dim W~_3 > dim(Wg2 2 - 1411) = g - 7, a contradiction. Thus dim W~_2 < g - 9. Since W~_2 \ (W~_ 3 + W, ) is not empty there exists a base point free pencil g~_~ on C. If it is primitive C has the Clifford sequence ( c , . . . ,9 - 4,9 - 3) since 9 - 4 > ~2 > c; in particular g > 3. So assume there is no primitive 9~ 2 on C. --Then the set W~_2 \ (W~_ 3 + W1) of dimension 9 - 6 is contained in W]_, - W1 whence dim Wff_1 >_ 9 - 7. We see that there is an at least (9 - 7)-dimensional set of base point free nets 9~-1 on C. Since Wg3 - W1 = ~ - (W]_ 2 + W1) has dimension at most 9 - 8 it follows that there are primitive 9~-1 on C, of Clifford index g - 5. Thus for 9 > 10 the three different values c,9 - 5,g - 3 belong then to the Clifford sequence of C. According to the Examples (1.1.2) and (1.2.7) we have shown that g = 2 and g >__ 10 imply c = 2. Leaving aside the case of a smooth plane sextic (cf. Example (2.3.2)) we thus conclude that C is 4-gonal. Consider 2941. If it is a gs3 C is bi-elliptic or 9 < 9. We therefore may assume that it is a 9s2 which is primitive if there is no 93 on C. But if C has a 9 3 it will be primitive because the existence of a 9s3 or a 9~0 again implies that C is bi-elliptic or 9 < 9. Hence we see that on a 4-gonal curve of genus 9 > 10 which is not bi-elliptic the series g~,g~ (resp. 93) and 9~-, are primitive whence g _> 3. | (2.3.2) E x a m p l e : A smooth plane sextic C has primitive length 2. In particular, since this curve has genus g = 10 we see that the genus bound in Proposition (2.3.1) is best possible. In fact, by a theorem of Max Noether (cf. [Ci], [Ital]), for smooth plane curves
248
COPPENS E T AL. of degree 6 we have =
X--
W 1
W~+i = x T W i
for
i=0,1,2.
W92 = (x+W3) U ( 2 x - W 3 ) = ( W ~ + W , ) U ( W 3 o - W , ) wg
=
if x E J(C) corresponds to the unique g~ on C. Assume there is a complete and base point free g~ on C. Then, by the base point free pencil trick (e.g. [ACGII],III, ex. B-4) dim(g~ + g~) + dim(g6 ' 2 - g~) , >_ 2 dim(g~) = 4 and dim(g~ - g~) = - 1 whence dim(g~ + g~) >_ 5, i.e. g~ + g~ = 2g~, and we obtain the contradiction g~ = g~. In the same way (using 2g~ instead of g~) we can handle with complete and base point free glr and g~ to obtain: w',
= =
w,'
+ w, + w,
=
Therefore, C has as primitive series only g~ and the g~-t = gl, i.e. C has the Clifford sequence (2, 7). 9 It would be very interesting to know the primitive length of all smooth plane curves (or at least the general ones). (2.4.1) P r o p o s i t i o n : A curve C of genus g > 19 cannot have primitive length 3. Proof: By Examples (1.1.2) and (1.2.7) wc may assume c _> 2. We distinguish several cases: 1) Claim: Let dim WJ_ 5 = g - 12. Then there exists a primitive series G on C such that g - 9 _< cliff(G) _< g - 6.
Proof of the claim: We havc dimWJ_ i = g - 2(i + 1) for 0 < i < 5, because dimWJ_ s = g - 12 = p ( g - 5,g, 1) and because dimW~ = p(d,g,r) implies p(d + i, g, r) g - l l . Note that dim W~_ 4 _< g - 13 since > would imply dim Wl_s >_ dim(Wg2_4 - W1) > g - 12, a contradiction. Thus we see that 14/']_a \ (W~_ 4 + W1) is now an at least
249
C O P P E N S E T AL. ( g - ll)-dimensional set representing base point free nets gg-3 2 of Clifford index g - 7. If none of these is primitive this set is contained in Wga- 2 - W1 whence dim Wga2 > g - 12. Since we have dim W~_ 3 a < g - 14 (otherwise dim WJ_~ > dim(Wga_3 - I,V2) > g - 12) we conclude then that W~_ 2 \ (W~a_3 + W1) has dimension at least g - 12 and is made up by base point free webs gg_~a of Clifford index 9 - 8. If none of these is primitive the same reasoning shows then that dim(}Vg4_~) < g - 15 and that W~_ 1 \ (W~_~ + W1) has dimension at least g - 13. But W~ - W~ = ~ - (W2_2 + W1) has dimension at most g - 14. Therefore, after all, there is a primitive g~-x on C, of Clifford index g - 9. _
2) Claim: Let g >_ 19. Then dim WJ_~ > g - 12 if and only if dim W~ > 1. Note that by this claim dim W~_ s > 9 - 12 implies We r 0 ([ACGtI], VII,ex. C) whence c < 5. Therefore, by 1), c >_ 6 implies the existence of a value m,g-9 19 and c>6.
Proof of the claim: Clearly, dim W~ > 1 implies dim }V~_5 > dim(Ws~ + }Vg-,a) > 9 - 12. T h e converse, for g _> 19, is a consequence of [M3], Theorem 3. 3) Assume that 3 < c < 5. Let r3 be the minimum of all d such that there is a g] on G' (i.e. r3 is the minimal degree of a web on C). Clearly, a 9,~ is complete and base point free, and we have (e.g. [M1], w r3 < 1 + 394A ~ whence the Clifford index of this minimal web is at most 9 - 6, for 9 > 16. T h e r e are three possibilities: (i) T h e minimal web is primitive and does not compute c. Then, c, r3 - 6 , n and 9 - 3 belong to the Clifford sequence of C where n = 9 - 4 or 9 - 5 (cf. the proof of Proposition (2.3.1)), and g > 4. (ii) The minimal web is primitive but computes e. According to [M1], Satz 4 this implies 9 < ~(c + 5)(c + 6), and for c < 5 we obtain the contradiction 9_ c we are done. So let cliff(g1) = c. T h e n (7 is birationally equivalent to a plane curve of degree d = c + 4 or is a double covering of a smooth plane curve of degree ~+4 2 " Let e = 5 . Then C is a plane curve of degree 9. If it has a primitive g~ we
250
C O P P E N S E T AL. are done. If not, all gs1 are obtained from nets of degree 9 which are finite in number ([A2], Tam. 7.1). Thus dim Ws~ = 1. Then the claim in 2) shows that dim WJ_5 = g - 12, and from the claim in 1) we see that s > 4. Let c = 4. If C is a smooth plane octic (g=21) then 5gs2 is the canonical series of C whence 2gs2 = g~s is primitive. For g < 20 or if C is a double covering of a smooth plane quartic there is a primitive g6~ on C, and dim(g~ + g6~) > dim(gs2 + g ~ ) + dim(g~ _ g l ) >_ 2dim(g~) = 4. Any g[4 (computing c = 4) or g~5 on C would be simple whence the contradiction g < 18. Thus gs~ + g~ = g~4 is a primitive series of Clifford index 6. The case c = 3 is impossible since C is then a plane septic whcnce g < 15. 4) Let c = 2, and assume that C is not bi-elliptic. Then C cannot have a g3 (otherwise C is trigonal or g < 12). C has a g4l, and the g~ and 2g~ = g~ are primitive. Consider 394~. It cannot be a g162or g[2. If it is a g~2 we have g < 15 unless C is trigonal (impossible) or a double covering of a genus 2 curve. If the latter does not happen 3g~ = g32 is primitive (a g~3 would imply g < lS again), and C has the numbers 2,4,6 and g - 3 in its Clifford sequence. 5) So we end up with a double covering C of a genus 2 curve. By 4), C has the Clifford sequence ( 2 , 4 , . . . , g - 3), and wc want to find another member of this sequence. We have W~ = W~)+4 + Wa-(2~+4) for 2r + 4 _< d < g - 4, and in [IIo] (cf. the appendix) it is stated that WJ_ 2 -fi W~_ 3 + 14/"1 and W2g_3 = W ~ 4 + W1 = W2s + W g - n . IIere we add the Claim: (i) W ; ~ = Ws2 + Wg_,0 (ii)
+ w._,
=
-
+
%_9)
Proof of the claim: For P E C we denote by P tile conjugate point of P on C with respect to the canonical automorphism of order 2 of C given by the 2-sheeted covering f of C over a curve of genus 2. (i) We have to show W~_~ = W~_3 + W1. So assume there is a complete and base point free gg-2 2 = [D[ on C. This net cannot be very ample since the relation 2g = (g - 3)(9 - 4) is impossible. Thus (cf. [ACGII], I, ex. D-l) there exist two points P1,P2 E C such that ID - P1 - P2[ corresponds to a point of WJ_4 = WJ + Wg-,0, i.e. IDI = b~ + P, + P2 + P3 + . . . + G - s l for some pencil 961 and some points P 3 , . . . , Pg-s r G. Suppose that our 9{ is base point free or of the form 9{ + P + P ( P E C), i.e. g~ = f 9 (g3)" 1 Then no two of the points P 1 , . . . , Pg-s are conjugate since
otherwise our g~-2 would correspond to a point of W~ + Wg-lo and so would not be base point free (for g > 11). Since the linear series [g~+Pl+...+Pj+-f,+...+-fjl = J'"(g3+j'+J')(J = 1, . .., g - 8) is base point free, P j is not a base point of l D + T l +. 9. + P J l = If . -(g3+jJWPJ+lW'"WPu-sl l+j~ 9
251
COPPENS ET AL. Therefore we have ]D + P , + . . . + P---jI~]D + P, + ... + -P.i-,[ + -Py, and we obtain h~ +-fix + . . . +-fij) > h~ +-ill + . . . +-fij-1). In particular, then, [ D + P 1 + . 9.+Pg-8[ corresponds to a point of (a-2)+(a-s)H]24(g-s) = W2g-10g-6 _. ~__ W3 which is impossible since Ws3 = 0. Assume that our 9sa is of the form g~ + P + Q (P, Q E C) where Q :~ P. Set Pa-r = P, Pa-6 = Q- Then [D[ = [g~ + Pa -4-... + Pa-6[, and under the g - 6 points P 1 , . . . , P g - 0 there is at most one pair of conjugate points (otherwise gs2 C [D[ again). If there is such a pair we are in the case already considered. If not, we obtain by the same reasoning as above that [D + P1 + ... + Pa-6t is a ~a~s which is impossible. This proves our assertion (i). (ii) For any choice of g - 5 points P 1 , . . . , P4, Q ~ . . . , Qj_-9 of C we have that L := ](P1 + . . . + P, + Q, + . . . + Q,-9) + ( P , + . . . + P 4 + Q, + . . . + Qg-9)l represents a point of W~9_1o "-~ = ~ - W~. Thus ]lfc L] is a gs~. Clearly, IP1 + . . . + P4 + P1 + ... + Pal also is a net of degree 8, h i say. Therefore, we may write Ih'c - (hs~ + Q, + . . . + Qa-9)[ = 19~ + Q1 + . . - + Qg-gl, i.e. we have - (Ws~ + Wo_0) C_ Ws2 + l,Va_0. Thus these (g - 7)-dimensional irreducible sets are equal and our claim is proved. Now it is easy to see that G has g ->- 4. Assume that W. q2- - 1 = W~ + Wg-9. Then we have W ~ 1 - W~ = (Ws~ - H q ) + Wg-0 C_ W~ + Wg-o = W6~ + W.9-s c_ WJ_a+W1. Consequently, since I,V~_2 r WJ_3+W1 , there are primitive g~-2 on G, of Clifford index g - 4. Thus we may assume that W~_, r W~ + Wg-9. By our claim, then, I,V3~-W1 = to- (Wff_2+W1) = ~ - (Ws2 +Wg_9) = Ws2 + Wg_9. Hence there is a primitive net g~-a on C, of Clifford index g - 5. So C has the Clifford sequence ( 2 , 4 , . . . ,n,g - 3) with g - 5 < n < g - 4. This proves our Proposition. [] (2.4.2) R e m a r k : Let G be a double covering of a genus 2 curve (9 > 11). Then it is not hard to see that
W~=W~+Wa_6 I,V~=W~+Wd_s W~ = W ;2,+4+Wd-(~+4)
for 6 < d < g - 4 for 8 < d < g - 2 for 2 r + 4 < d < g
1
and
r_>3.
In particular, C has primitive length 4 or 5 since for d > g - 3 the primitive series g~ on C are among g.9-3, 1 gg-2,ga-1 1 1 and ga-a" 2 [] In this w we have proved that a curve C of genus g > 19 and primitive length < 3 is hyper- or bi-elliptic; so in particular C is a double covering. This will be generalized in the next w As we have already seen, in the discussion of the existence of primitive series on C dimension-statements for the varieties of special divisors, W[l C_ J(C), are indispensable. Therefore, we will begin with such statements.
252
COPPENSET
3
AL.
D i m e n s i o n t h e o r e m s for WS and d o u b l e coverings
We first present some dimension theorems for W~. Recall that dim W~ _< d - 2 r if W~7 ~ 0 and r > d - g . For r = 1 we have the basic
(3.1.1) T h e o r e m g- 1 -j
(Coppens [Co2]): /jr dim W~ = d - 2 - j f o r j + 3 < d < (j >>_O) a n d g > (2j + 1)(j + 1) then diml4/)+ a = 1. 9
(3.1.2) C o r o l l a r y : Let n e N, g >_ ( 2 n - 3 ) ( n - 1 ) (and i f n < 2 letg > 2 n - l ) . Assume that dim Wn~+I 1 < 1. Then W~ is equidimensional of the minimal possible dimension p(d,g, 1) = 2d - 2 - g for all d such that g - n < d < g. Proof: Write dimW~ = d - 2 - j and assume dimW~ > 2 d - l - g . Then this implies d < g - 1 - j . On the other hand, d _> g - n + 1. Combining these two inequalities we obtain n k j + 2. Note that 2 d - 1 - g > 2 ( g - n + l ) 1-g = g-2n+l > 0 by our assumption on g. Thus we have W~ =~ qJ. So d > gonality(C). By our assumption, gonality(C) > n + 1. Hence d k n + l >_ j + 3 . By the theorem, then, dim 141]+a = 1, and we obtain the contradiction dim W~+ 1 > 1. 9 (3.1.3) C o r o l l a r y : Under the assumptions of Corollary (3.1.2) we have complete and base point free pencils gg,gg-l,'",gg+~-,~ 1 1 1 on C, f o r n > 2. 9 Recall that dimW~-~ _> dilnW~ + 1. Thus Theorem (3.1.1) implies: If dimW~ = d - 2 r - j (r > 1,j k 0) for d ( 2 j + l ) ( j + l ) then dim W)+ a _> 1. For j < r this applies to all d _< g - 1. Itowever, for j < r we can be much more precise: We have the following remarkable generalization of II. Martens' well-known dimension theorem ([ACGIt, iV, Theorem 5.1): (3.2.1) T h e o r e m :
Let d < g - 1 and O < j < r. Then dim W~ = d - 2 r - j implies that C is a double covering of a curve of genus j unless d = g - 1 = 3 r - 1 and j = r - 1 > 1 in which case C can also be an extremal curve of degree 3r - 1 in ]W.
Proof: For j = 0 the theorem is nothing but tI. Martens' theorem. So let j > 1. In particular, C is not hyperelliptic. Let Z be an irreducible component of W~ of maximal dimension. If a general element of Z corresponds to a simple series g~ on C (r > 2) we can apply the Accola-Griffiths-Harris-theorem ([ACGH], IV, ex. E). Thus either 2g,~ is the canonical series of C, or d - 2r - j = dim Z < d - 3r. In the latter case we obtain the contradiction j >_ r. Hence g~ is a theta characteristic which implies d i m Z = 0. Then d - 2 r = j < r, i.e. d < 3r, and it follows from Castelnuovo's genus bound for curves in Pr that the g~ embeds C in P" as an extremal curve of degree d = 3r - 1. (Cf. [CM], Prop. 2.4.1.)
253
C O P P E N S E T AL. Hence we may assume that a general element of Z corresponds to a comr It will give us a composition of morphisms C ~ C' --4 pounded gd" l' pr with 1 < n := deg f I d, and f ' is associated to a complete g~ on the (smooth) curve n
C ~. We distinguish two cases: (i) Let the induced series g~ on C' be nonspecial. This implies that C' ha~ n
genus g' = -~ - r . Let g' = 0. Then n > 3, d = nr, Z C rW~, and dimW~ > d - 2 r - j = (n-2)r-j. Since dimW~ < n - 2 (recall that C is not hyperelliptic) we obtain the contradiction j > (n - 2)(r - 1) > r - 1. Thus g' > 0. Then there are only finitely many possibilities for coverings f of degree n (up to automorphisms), by de Franchis' thcorem. Thercfore, the induced map l9 W~(C') = J(C') ~ Z n
is dominant, and we have d-2r-j=dimZ i.e.(n-1)d
= =
dimJ(C')=g'= n(r+j).
d_r, n
Since d > 2 r + j (note that 0 _< dimW~ = d - 2 r - j ) wc get ( n - 2 ) r Thus n = 2, i.e. C i s a double cover, and d = 2 ( 7 - + j ) , g ' = ~d - r predicted.
_< j < r. = j , as
(ii) We still have to exclude the case that the induced series g~ on C' is n
special. If so, we clearly have g ~ >_ 2, and we have an induced dominant map f* from some irreducible component of W~(C') to Z. Thus, by the It. Martens' r8
theorem (on C'), d-2r-j=dimZ
_< d i m } V ~ ( C ' ) < _ - a - 2 r , n
n
i.e.(n-1)d
_< nj.
Since d _> 2r + j we obtain the contradiction 2(n - 1)r _< j < r. The curves described in Theorem (3.2.1) are easily seen to be of an even gonality. Therefore we obtain the (3.2.2) C o r o l l a r y : On a curve C of odd gonality we have dim W~ _< d - 3r ford 3r we h&vc dim W~r = 0, i.e. for j = r Theorem (3.2.1) is false. For j = 1 Theorem (3.2.1) reduces to a refinement of Mumford's theorem studied in [ACGH], IV, ex. D. Note that our proof works without first reducing to pencils. By Corolla.ry (3.2.2), on a curve C of odd gonality the theorem of Clifford may
254
C O P P E N S E T AL. be improved ([CM], Cor. (2.4.3)): We have d > 3r for a g~ (d < g - 1) on C.
It is the aim of this w to apply these two dimension theorems (3.1.1) and (3.2.1) to give a characterization of double coverings by looking at the length of the Clifford sequence resp. of the "gonality sequence" (eft (3.5)) of the curve. To begin with, we want to have an analogue to Corollary (3.1.3) for primitive pencils. To do this, we need an appropriate analogue of Corollary (3.1.2) for nets. Note that a straightforward application of Theorem (3.1.1) yields: If n E N, g > (2n - 1)n and dimW~+ 2 < 1 then dim[,V~ _< 2 d - 5 - g for g - n < d < g. This is too weak, however. To succeed, we have to prove our third dimension theorem: (3.3.1) T h e o r e m : Let n E N , g dimW~ dim Z >_ g - 2m - 5 = (g - m - 2) - 2 - (m + 1), and from Theorem (3.1.1) (for j < m + 1 < n) we obtain dim I/Vn1q - 3 > 1, a contradiction to our hypothesis. ---So z corresponds to a base point free net g~_m(z). If it is not simple we again haveZ-W1 C 1 - W,-,,,-2 + W1 leading to the same contradiction to Theorem (3.1.1) as before. Finally, if g~_m(z) is very ample the curve C (being now a smooth plane curve of degree g - m) must have gonality k = g - m - 1 >_ g - n such that dim W~ = 1. Since dim W2+ 3 < 1 it follows k _> n + 4. Adding up both inequalities for k we get 2k >_ 9 + 4 which cannot hold for the gonality of any curve (e.g., [ACGIt], V, Thin. 1.1). This proves the claim. We see that the plane model of C of degree g - m associated to g~_m(z) has a singular point. In particular, Z C l,V~_m_2 + Vs. So there exists an irreducible component Y of W~_m_ 2 such that Z C_ Y + W2. Let y be a general element of Y. First we note that Z ~ Y + W2: Otherwise we would have dim [ggX_m_2(y) + P1 + Psi > 1 = dim [g~-,,-2(Y)l for a general choice of two points P1,P2 on C (such that ]P1 + Psi corresponds to a general element of W2). But this is impossible si~lce dim II(v - g~-m-2(Y)l = m + 2 > O. We may write d i m Y = g - 2 m - 7 + s w i t h s >_ 0. If s = 0 w c c l e a r l y h a v c Z-- y+w2 which is excluded. I f s > 2 we have d i m W 1 > dimY > --
255
g - m - 2
--
-
-
C O P P E N S E T AL. g - 2 m - 5 , and we get the same contradiction to Theorem (3.1.1) as before. So 8 = 1, i.e. d i m Y = g - 2 m - 6 . (Then d i m Z = g - 2 m - 5 since Z 7~ Y + W 2 . ) Consider the natural map f : Y x W2 ~ Y + W2. We have Z C_ Im ( f ) , hence d i m f - ~ ( Z ) _> g - 2 m - 5 . Let p~ : Y x W2 ~ Y be the projection onto the first factor. Since d i m Y = g - 2 m - 6 we conclude dim(p[~ (y)N f -~ ( Z) ) >_ 1 (y E Y general), i.e. for a general element y E Y there exists a one-dimensional family bye, say, of effective divisors E of C of degree 2 satisfying 1 -- dim
Ig$-m- (y)l
g-3m-5, --2 --
and we get d i m
> dim(W
A
- W , , ) _> g - 2 m - 5. T h i s once more
contradicts T h e o r e m (3.1.1). So let d < m. We have already observed that [Kc - g~-m-2(Y) -- D[ is a base point free and compounded Yg+m-d-m+2thus defining a non-birational morphism h : C ~ p,,,+2 Let C' be the normalization of the curve h(C) and let h also denote the induced covering map C ~ C '. Finally, let k > 2 be the degree of the covering h.
Claim 2: C' is not rational. To prove this, let us assume that C' is a rational curve. Then h gives us a g~ on C, and IKo - g~_,._~(y) - D I = (m + 2 M . We have dim IKc - g~-,,,-2(Y) - D - gJl = m + 1, and we see that
256
C O P P E N S E T AL. = dim tKc 1. By the Riemann-Rochtheorem, then dim[g~_,,,_2(y ) + g~[ = dimlg~_m_2(y)[ + k - 1 = k. So the dim IKo
k complete pencil g~, corresponds to a point x E W2 such that x + y E W~_,,_2+ k.
Let us consider tile incidence correspondence I C W2 x Y defined by: (x, Y) E I k if and only if x + y E W~_,,,_2+ k. Let pl : I ~ W2 and p2 : I --4 Y be k the projections and q = pl + p2 : I ~ W~_,,_2+ k. We observed that p2 is dominant, and so d i m I > If dim W~-m-2+k k - g-2m-6. > g - 2m - 3 - k then dim W g1- - m - 1 > 9 k d,m(W~_,,,_~+ k W k , ) > 9 2 m 3 = (g-m-1)-2-m, and ~ by Theorem (3.1.1) (for j _< m < n) we get the contradiction dim W2+ 1 2 _> 1. So, dim W~-m-2+k k < 9 -- 2m -- 3 -- k. This implies that the non-empty fibres of q have dimension at least k - 3. Let Io be a general non-empty fibre of q. If (x~,y~) E Io and (x2,92) E I0 then x~ + y~ = x2 + Y2. Ilence x~ = x2 if and only if 91 = 92. Therefore, the restriction of pl to I0 is injeetive, and we obtain dimpl(10) = diml0 > k - 3 . In particular, dimW2 > k - 3 . Ifk < 9-2 then Theorem (3.1.1) implies dimW4~ > 1, a contradiction to dim W~+z 1 < 1 (note that n _> 1). So k > g - 1. But t h e n g + n > g+m > g+m-d = deg(Kc - g~-~-2(Y) - D) = (m + 2)k > 2k > 29 - 2, i.e. n > g - 1. tIence 1 W~+ 3 = J ( C ) of dimension zero, and g = 0. This is excluded, and Claim 2 is thereby proved. By Claim 2, C has an irrational involution of order k > 2. By the de FranchisTamme-Thcorem, the number of these involutions (k fixed) on C is finite. Therefore, there is a morphism h : C --~ C' of degree k onto a curve C' of genus g' > 0 and an irreducible closed subset Y' of W~+.... e (C') such that k
~o-Y C h ' ( Y ' ) + Wd. Hence, g' > d i m Y ' >_ dimh*(Y') >_ d i m Y - d . Now recall that d < m. This i m p l i e s g ' - i > dimY-m =9-3m-6 > 9 - 3(n + 1). On the other hand, according to the Riemann-IIurwitz-relation we have 2 9 - 2 > k ( 2 9 ' - 2 ) > 2 ( 2 9 '-2),i.e. 9'-1 < ~ Thus we o b t a i n g < 6 ( n + l ) . S i n c e 0 < d < m < n we have n _> 2, and our genus bound g >_ 2(n + 1) 2 gives a contradiction which finally proves our theorem. 9 (3.3.2) C o r o l l a r y : Under the assumptions of Theorem (3.3.1) we have primitive pencils g ~ - l , " . ,g~-, on C. In particular, g > n + 1. P r o o f : Combine Corollary (3.1.2) and Theorem (3.3.1). For g, observe that tile 9~ oll tile k-gonM curve C is another primitive pencil. 9 We are now in a position to prove our main result for tile primitive length g: (,3.4.1) T h e o r e m : Let go E N and C be a curve of primitive length g < go and genus g >_ ~(go + 3) 3. Then C is k-gonal with k _< go + 3, and for k > 2 every pencil g~ on C is compounded of an irrational involution of order 2.
257
COPPENS E T AL. P r o o f : Our assumption on g implies g _> 2(go+l) 2. Taking n = g0 in Corollary (3.3.2) we obtain dim W~o+a 1 > 1 from g < go. Thus k _< go + 3, and it remains to show that tim map associated to a g~ factors through a double covering C - - , C'. Let k > 3 and [D[ = g~. By Lemma (1.2.3) all ]rD] are primitive for r = 1 , . . . , tk(k-~)J, and cliff(sD) _< cliff(rD) for s _< r. If wc always have strict inequality here we obtain (by recalling that there also is a primitive pencil in degree g - 1) O
(1)
g_> [ 2k-~
-
2
2g - 2
~)]+1
> (go+3)(go+2) >g~
by our choice of g. This contradiction shows that there is a minimal r _< .2~-2 k(k-1) such that (2)
cliff((r - 1)D) = cliff(rD).
Then all 18D] such that 8 < r (and also some pencil g~-a) contribute to g, and so we have g0 > s > r. I f h ~ = r + 1 we also have h ~ -- r contradicting (2). Thus h~ > r + 1. In particular, [rD[ cannot be simple since otherwise g < {k(k - 1)r by Castelnuovo's genus bound contradicting our choice of r. Hence the map associated to [rD[ is not birational. Lct E be the normalization of the image curve of this map, and let c : C ~ E be the induced map. Then e factors through a morphism f : C --~ C' onto a smooth curve C t C [D[ f
/" P'
N C'
e ~
%
J E
such that f is not birational and admits no nontrivial factoring. Let m =
deg(e) and n = deg(f). Clearly, [D[ = e*([Dm[), [rD[ = e*([rDm[) and [(r - 1)D[ = e*([(r - 1)Din[) where [Dm[ is a pencil of degree •m on E and h~ = h~ (eg. [MI], Lemma 3), h~ - 1)Din) = h~ - I)D). Therefore, there is a pencil [D,[ of dcgree _k on C' such that [D[ = f*([D,,[), h~ = h~ and h~ - 1 ) n , ) = h~ - 1)D). By (2), then, cliff((r - 1)D,) - cliff(rD,) = 2(h~ - h~ - 1)D)) - ~- = k - k > 0 since n _> 2. Let g' be the genus of C'. If 2r~ < g' we certainly have h~(2rD~) >_ 1, and then [ELMS], Lemma 3.1 (cf. (0.2)) gives a contradiction to our just obtained strict inequality of Clifford indices. So g~ < 2r~. Recall that the map f : C ~ C' does not factor. Therefore, any meromorphic function on C of order d < ~ + 1 has to be contained in the function field
258
C O P P E N S E T AL. of C' ([M1], w [ACGH], VIII, ex. C-I). Consequently, for d < ~ + 1 any base point free 9~ on C is induced by C'. In particular, any base point d ~_g~ free and nonspeciM gd' on C' - such a series exists for d' > J - induces a d w_9 primitive series 9.a, on C if n d < ~ since there cannot be a complete and base point free Y,a,+~ A'-y+I on C. We have chff(g~a, 9 a'-g' ) = (n - 2)d' + 2g'. Assume that n > 2. Then all these series contribute to g, and so we have g >- - u - " 2 a ' - 1 Recallingg' < 2 r ~ , r < g o , n < k and k < go + 3 we obtain from our hypothesis g _> ~(go + 3) 3 the contradiction g > go. So wc must have ~
n ( n - 1 )
"
- -
--
- -
- -
n = 2, and our g~ is compounded of an irrational involution of order n = 2
and genus 9' 4, and we have L < [z~_!~] by Clifford's theorem. Clearly, d, < rdx, and so L >_ [z~fl] if G is k-gonal. In fact, for trigonal curves we always have d~ = 3r, i.e. L = [u._~], and, consequently, L does not distinguish among trigonal curves of genus g. The 4-gonM curves of Example (1.3.2) are also not separated by L. So this invarinnt seems to be less sensitive on Mg.k than the primitive length g. (3.5.1) E x a m p l e : Let C be the general 4-gonal curve of genus 8. Then C has a 9r2 corresponding to Segre's plane model of U ([AC]). Thus L = 2 exceeds MaX([vf~] - 1, [u-~])which is only 1 for g = 8, k = 4. 9 (3.5.2) E x a m p l e : Let C be an extremal curve of degree d > 2r in ]W(r >__2; cf. [ACGH], III, w Then C has a unique and very ample g~. We will give some bounds f o r L , for this curve. L e t r , := 8 9 (nEN) and let m be an integer such that a_~ < m < ~-~ + 1. According to a theorem
259
C O P P E N S E T AL. of Ciliberto ([Ci],2.11), for L > rm a ggL is nothing but mg~a whence m = and L = dim(ragS) = r,,. Assume that m # a-~d~. Then we nmst have L < r,,. Put t := [a_.~]. Since td < g - 1 and dim(tg~) = rt we conclude that L > r,. We thus have proved that rt ~3 are double coverings. More precisely, (3.5.3) P r o p o s i t i o n : Let eo E iv, u {0}. If L = [z~] _ go and if g > 6(g0 + 1) then C is a double coverin 9 of a curve of genus go. Conversely, if C is a double covering of a curve C' of genus go and if g > 690-1 then L = [ a . ~ ] _ go. (Note that L does not depend on the peculiar features of the covered curve C'). P r o o f : Let L = [a_~] _ Q. Then tim bound for g implies dL+ 2 _ ~ 2 - 9 ' r_> ~:22 - 9 '
if 9 if g
is odd is even,
and so L > [g_~_l]_ 9'. Tiffs proves 9' = Q. Conversely, let C be a double cover of seen that we then have L >__ [9-1 2 ] -go. and Corollary (2.2.2) implies L = dL2 -g = 6Q - 1. Then L > a [ a ~ ] - go = 2Q the contradiction L = ~ - go < 2Q - 1.
a curve of genus go. We have just For g > 6go we thus get L > 2Q, go _< [a_~] _ g0. So let g = 6Q or 1. If L > 2go Corollary (2.2.2) gives Thus L 2Q - 1 = [a_~A] _ Q. 9
(3.5.4) R e m a r k : Another upper-semicontinuous invariant of curves C of genus 9 possibly exhibiting some (rather distinguished) double covers as being "most special" is the order IAutC] of the group of all automorphisms of C. It is already implicitly stated in Hurwitz' classical memoir [Hu] that IAutC[ > 15(9 - 1) implies that ]AutC] is even whence C obviously is a double cover. Unfortunately, there is no general formula known to compute the number A(g ) := Max([AutC] : C of genus g). IIowever, it is known (e.g. [ACGIt],I, ex.F) that 84(9 - 1) > A(g) > 8(9 + 1) and that A(g ) attains the minimal value 8(g + 1) and also the maximal value 84(g - 1) for infinitely many g. One should expect that A(g) is always even. l
260
C O P P E N S E T AL. Appendix In this appendix we want to prove the L e m m a : Let C be a double cover of a genus-2-curve C t of genus g > 13. Then C has a complete and base point free pencil gg-21 of degree g - 2. We used this result in the proof of Proposition (2.4.1). It is the main result of [Ito] but in our opinion there is a gap in the last part of Itoriuchi's proof: He obtains a singular plane model P of degree g of C with a singular point s of multiplicity at most g - 4. By the well-known formula for tile gcomctric genus of a singular plane curve the plane model P must have further singularities. But unfortunately they all could be virtual points of F, i.e. all singularities of P different from s could be infinitely near singular points lying over s. Then Horiuchi's projection argument to obtain a complete and base point free pencil of degree at most g - 2 on C does not work. (By the way, the same objection applies to Shokurov's proof [Sh] of the existence of a complete and base point free pencil of degree g - 1 on a bi-elliptic curve, i.e. the reducibility of W~_ 1 for such a curve.) To begin with, let f : C ~ C' be the double covering map. (Note that for g _> 10 C cannot be bi-elliptic and f is the only involution of order and genus 2, according to [ACGH], VIII, ex. C-1.) Consider the dual series IKc-g~4[ = g~g-6g-4 of the unique 941 = f*(llfc, D on C. Since W62 = 0 we have dim [g4~ + P + Q[ = = 1 for any choice of points P, Q on C. Therefore, dim [Kc - g~ - P - Q[ = dim [ K c - g ~ ] - 2 , and we see that ] K c - g ] [ is very ample. If we identify C via ]Kc - 941[ with its image curve of degree 2g - 6 in ~g-4 then, by the uniform position theorem, any g - 6 points P 1 , . . . , Pg-6 of a general hyperplane section of C span a Pg-r such that the projection C ~ It~ with center this pg-r gives us a (singular) plane model F of C, of degree (29 - 6) - (g - 6) = 9. Let P 1 , . . . ,Pg-6 be the conjugate points of P I , . . . , Pg-6. Since the series G :=
](Uc - g~ - P1
-...-
P.-6)
.
P,.
.... .
T.-61 .
]I(c
f
" (g "-~-4)l
is a g~ on C we see that F has a singular point s of multiplicity at lcast g - 6 (corresponding to the "linear span" of the points T ~ , . . . ,Pg-6 in F'2). Next we show that we may assume that this point has multiplicity cxactly g - 6. In fact, let it have multiplicity g - 5 or more. Then G is a 96I with base points, i.e. G = g~ + P + Q with P, Q E C (note there is no base point free g~ on C), and the multiplicity of s is g - 4 (associated to P~ + . . . + P g - 6 + P + Q ) . Since IKc - G + P + P] = f ' ( g - a 5) is base point free we have dim [Kc - G + P + P [ = dim [Kc - G + P] + 1, and since P is a base point of G we have dim [Kc - G + PI = dim IKc - G[ + 1. Thus dim IG- P - PI = dim G, and so the two base points P,Q of G a r e P , P , i.e. we obtain Q = P and G = g~ + p + ~ . In particular, then, G varies (viewed on J ( C ) ) in an only
261
COPPENS ET AL. one-dimensional set. But the set gc - f*(~c, + Wg_6(C')) has dimension dim J ( C ~) = 2. So, for a general choice of P 1 , . . . ,Pg-6 on C, the series G is base point free, and our singularity 8 has multiplicity g - 6. We want to show that s is an ordinary singularity. To see this it is enough to prove t h a t ]G - Pi - Pj[ = 0 for 1 _< i < j < g - 6. We use a mono d r o m y argument. Consider the correspondence Z C C a-6 • C (4) where C (4) denotes the set of effective divisors on C of degree 4 and Z is defined by: ( P 1 , . . . , P g - 6 , D) E Z for P 1 , . . . , P g - 6 E C and D E C (4) if and only if D E I(Kc-g~-PI-...-Pg-6--fil-...-P--a-6)-P~-P21. Let p : Z --* C g-6 be the projection into the first factor. Assume that p is surjective. For a general choice of P ~ , . . . , Pg_6 on C we have that G = ]lfc-g~ -P1 -...-Pg-6-P~-...-Pg-6l is a base point free pencil and so ]G - P1] is a fixed effective divisor of degree 5, F say. Let a be a p e r m u t a t i o n of { 2 , 3 , . . . , g - 6}. Since we assume p(Z) = C g-6 we clearly have ( P I , P , ( 2 ) , . . - , P ~ ( g - 6 ) ) E p(Z), and so IG - P l - Po(~)l = IF - Po(2)] is not empty. Thus wc obtain P~(2) _< F, and we see t h a t Pz + . . . + Pg-6 _< F which implies g - 7 < 5. But we have g > 13. So this contradiction shows p(Z) ~ Cg-6. Since C g-6 is irreducible, for a general choice of ( P ~ , . . . , Pg-6) E C g-6 and for an arbitrary permutation a of { 1 , . . . ,g - 6} we have (Po(1),..., Po(a-6)) r I,(Z), and we are done. So there lic no singularities of F above s. If s is the only singularity of P we obtain fi'om the genus formula of a plane curve that F has the geometric genus g = 89 - 1)(g - 2) - (g - 6)(g - 7)) = 5g - 10 which cannot hold. Hence there must be another singular point s ~on P. Looking at the line joining s and 8~ Bezout's theorem says that s ~ has multiplicity at most 6. Thus the pencil of lines through s ~ gives us a complete and base point free g~ on C such that g - 6 < d < g - 2. However, 6 < d _< g - 4 is impossible (cf. Corollary (2.2.2)), and we see t h a t for g > 13 we can only have d = g - 3 or d = g - 2 . By tIoriuchi's l e m m a ([Ho]), a complete and base point free pencil of degree g - 3 implies the existence of such a pencil of degree g - 2. This proves the lemma.
References
[hl]
R.D.M. Accola: Plane models for l~iemann surfaces admitting certain halfcanonical linear series. Part I. Proc. of the 1978 Stony Brook Conference "Riemann surfaces and related topics". Princeton Univ. Press 1980
[i2]
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[ACGIf] E. Arbarello, M. Cornalba, P.A. Grittiths, J. IIarris: Geometry of algebraic c u r v e s . I. Springer 1985
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C O P P E N S E T AL. [B] E. Ballico: A remark on linear series on general k-gonal curves. Boll. UMI (7), 3-A (1989), 195-197 [Ch] E.B. Christoffel: 0ber die kanonische Form der Itiemannsdmn Integrale erster Gattung. Ann. di mat. (2), 9 (1878), 240-301 [Ci] C. Ciliberto: Alcune applicazioni di un classico procedimento di Castelnuovo. Seminari di Geometria 1982-1983, Univ. di Bologna 1983 [Col] M. Coppens: One-dimensional linear systems of type II on smooth curves. PhD-thesis Univ. Utrecht 1983 [Co2] M. Coppens: Some remarks on the schemes W~. Ann. di Mat. (4), 157 (1990), 183-197 [CM] M. Coppens, G. Martens: Secant spaces and Clifford's theorem. Composito Math. 78 (1991), 193-212 [EC] F. Enriques, O. Chisini: Lezioni sulla teoria geometrica della equationi e delle funzioni algebriche. Vol. III. Bologna 1924 [ELMS] D. Eisenbud, It. Lange, G. Martens, F.-O. Schreyer: The Clifford dimension of a projective curve. Compositio Math. 72 (1989), 173-204 [tIal] R. tlartshorne: Generalized divisors on Gorenstein curves and a theorem of Noether. J. Math. Kyoto Univ. 26 (1986), 375-386 [Ila2] P~. IIartshorne: Algebraic geometry. Springer 1977 [IIo] R. Horiuchi: Meromorphic functions on 2-hyperelliptic l~iemann surfaces. Japan. J. Math. 7(1981), 301-306 [Hu] A. llurwitz: Uber algebraische Gebilde mit eindeutigen Transformationen in sidl. Math. Ann..[1 (1893), 403-442 [J] F. Jongmans: Observations compl~mentaJres sur les s~ries sp~ciales des courbes alg~briques. Acad. Roy. Belgique, Bull. Cl. Sci. V 36 (1950), 128-137 [K] C. Keem: On the variety of special linear systems on an algebraic curve. Math. Ann. 288 (1990), 309-322 [M1] G. Martens: Funktionen von vorgegehener Ordnung auf komplexen Kurven. J. reine angew. Math. 320 (1980), 68-85 [M2] G. Martens: 0ber den Clifford-Index algebraisdmr Kurven. J. reine angew. Math. 336 (1982), 83-90 [M3] G. Martens: On dimension theorems of the varieties of speciM divisors on a curve. Math. Ann. 267 (1984), 279-288 iM4] G. Martens: On curves on K3 surfaces. In: Lect. Notes Math. 1389 (1989), 174-182
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COPPENS E T AL. [MS] G. Martens, F.-O. Schreyer: Line bundles ond syzygies of trigonal curves. Abh. Math. Sere. Univ. Hamburg 56 (1986), 169-189 [Sc] F.-O. Schreyer: Syzygies of curves with special pencils. PhD-thesis Brandeis Univ. 1983 [Sh] V.V. Shokurov: Distinguishing Prymians from Jacobians. Invent. math. 65 (1981), 209-219 Marc Coppens Katholieke Industriele Hogeschool der Kempen Campus H. I. Kempen Kleinhoefstraat 4 B - 2440 Geel Belgium (related to the University at Leuven, Celestijnenlaan 200, B-3030 Leuven, as a research fellow)
Changl[o Keem Department of Mathematics College of Natural Sciences Seoul National University Seoul, 151 - 742 Korea
Gerriet Martens Mathematisches Institut Universits Erlangen - Nilrnberg Bismarckstr. 1 1/2 D - 8520 Erlangen Germany
(Received May 14, 1992)
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