Varieties of Minimal Rational Tangents and Congruences of Lines - A

VMRT and congruences of lines G. Occhetta VMRT Definition Properties Examples Ir(reducibility)

Fano bundles Why Fano bundles? A classification theorem

Varieties of Minimal Rational Tangents and Congruences of Lines A classical counterexample to a modern conjecture

Congruences Generalities Special congruences Severi varieties Trisecants to projected Severi

G. Occhetta joint work with R. Muñoz and L.E. Solá Conde

MADRID, December 2012

VMRT and congruences of lines

Outline

G. Occhetta VMRT Definition Properties Examples Ir(reducibility)



1 Variety of Minimal Rational Tangents


Outline





Definition


Outline





Definition Properties


Outline





Definition Properties Examples


Outline





Definition Properties Examples Ir(reducibility)


Outline



Congruences



Generalities Special congruences Severi varieties Trisecants to projected Severi

2 Interlude: Fano bundles


Outline



Congruences





Why Fano bundles?


Outline



Congruences





Why Fano bundles? A classification theorem


Outline



Congruences





Why Fano bundles? A classification theorem 3 Congruences of lines


Outline



Congruences






Generalities


Outline



Congruences






Generalities Special congruences


Outline



Congruences






Generalities Special congruences Severi varieties


Outline



Congruences







VMRT and congruences of lines G. Occhetta VMRT

VMRT Families of rational curves




X smooth complex projective variety of dimension n






X smooth complex projective variety of dimension n Hom(P1 , X) scheme parametrizing maps f : P1 → X






X smooth complex projective variety of dimension n Hom(P1 , X) scheme parametrizing maps f : P1 → X Hombir (P1 , X) ⊂ Hom(P1 , X) open subset






X smooth complex projective variety of dimension n Hom(P1 , X) scheme parametrizing maps f : P1 → X Hombir (P1 , X) ⊂ Hom(P1 , X) open subset RatCurvesn (X) quotient of Homnbir (P1 , X) by Aut(P1 )






X smooth complex projective variety of dimension n Hom(P1 , X) scheme parametrizing maps f : P1 → X Hombir (P1 , X) ⊂ Hom(P1 , X) open subset RatCurvesn (X) quotient of Homnbir (P1 , X) by Aut(P1 ) Family of rational curves: V ⊂ RatCurvesn (X) irreducible component


VMRT

G. Occhetta

Families of rational curves

VMRT Definition Properties Examples Ir(reducibility)


Congruences Generalities Special congruences


Severi varieties Trisecants to projected Severi

U π

V

V is dominating if i(U) = X;

i

/X


VMRT

G. Occhetta

Families of rational curves





Severi varieties Trisecants to projected Severi

U

i

/X

π

V

V is dominating if i(U) = X; ex := π(i−1 (x)) is proper for a general x in Locus(V). V is minimal if V ex is smooth. For a general x ∈ X the normalization Vx of V


VMRT Definition




Given a minimal dominating family V of rational curves on X and a general point x ∈ X the rational tangent map is


VMRT

G. Occhetta

Definition



Congruences Generalities

Given a minimal dominating family V of rational curves on X and a general point x ∈ X the rational tangent map is τx : Vx

/ P(TX |∨ x)

Special congruences Severi varieties Trisecants to projected Severi

` → P(T` |∨ x) associating to a curve through x its tangent direction at x.


VMRT

G. Occhetta

Definition




Given a minimal dominating family V of rational curves on X and a general point x ∈ X the rational tangent map is τx : Vx

/ P(TX |∨ x)


` → P(T` |∨ x) associating to a curve through x its tangent direction at x. We define the Variety of Minimal Rational Tangents to be Cx = τx (Vx )


VMRT

G. Occhetta

Properties


Fano bundles

The subvariety


Congruences

C := closure of

[

Cx ⊂ P(TX∨ )

generalx∈X


is called the total variety of minimal rational tangents of V.


VMRT

G. Occhetta

Properties


Fano bundles

The subvariety


C := closure of

Congruences

[

Cx ⊂ P(TX∨ )

generalx∈X



Trisecants to projected Severi

Theorem • (Kebekus) τx : Vx → Cx is a finite morphism.


VMRT

G. Occhetta

Properties


Fano bundles

The subvariety


C := closure of

Congruences

[

Cx ⊂ P(TX∨ )

generalx∈X




Theorem • (Kebekus) τx : Vx → Cx is a finite morphism. • (Hwang-Mok) τx : Vx → Cx is birational.


VMRT

G. Occhetta

Properties


Fano bundles

The subvariety


C := closure of

Congruences

[

Cx ⊂ P(TX∨ )

generalx∈X




Theorem • (Kebekus) τx : Vx → Cx is a finite morphism. • (Hwang-Mok) τx : Vx → Cx is birational. • Thus τx : Vx → Cx is the normalization.


VMRT

G. Occhetta

Prime Fanos




The VMRT is defined for any uniruled variety.


VMRT

G. Occhetta

Prime Fanos


Fano bundles




X is a Fano manifold if −KX is ample.


VMRT

G. Occhetta

Prime Fanos


Fano bundles





Theorem (Mori 1979) Fano manifolds are uniruled.


VMRT

G. Occhetta

Prime Fanos


Fano bundles





Theorem (Mori 1979) Fano manifolds are uniruled. A prime Fano manifold is a Fano manifold such that Pic(X) ' ZhHX i. The index rX of X is the largest integer such that −KX ∼ rX HX


VMRT

G. Occhetta

Properties

VMRT Definition Properties Examples

A rational curve f : P1 → C parametrized by V is called standard if

Ir(reducibility)



f ∗ TX ' OP1 (2) ⊕ OP1 (1)⊕d ⊕ OP⊕n−d−1 . 1 where d = −KX · C.


VMRT

G. Occhetta

Properties



Ir(reducibility)



f ∗ TX ' OP1 (2) ⊕ OP1 (1)⊕d ⊕ OP⊕n−d−1 . 1 where d = −KX · C. The tangent morphism is immersive at p ∈ Vx if and only if i(π −1 (p)) is a standard rational curve.


VMRT

G. Occhetta

Properties



Ir(reducibility)



f ∗ TX ' OP1 (2) ⊕ OP1 (1)⊕d ⊕ OP⊕n−d−1 . 1 where d = −KX · C. The tangent morphism is immersive at p ∈ Vx if and only if i(π −1 (p)) is a standard rational curve. Proposition (Hwang)


VMRT

G. Occhetta

Properties



Ir(reducibility)

f ∗ TX ' OP1 (2) ⊕ OP1 (1)⊕d ⊕ OP⊕n−d−1 . 1



where d = −KX · C. The tangent morphism is immersive at p ∈ Vx if and only if i(π −1 (p)) is a standard rational curve. Proposition (Hwang) 1

Assume that X ⊂ PN , and that V is a family of lines. Then, for a general x ∈ X, τx is an embedding and Cx is smooth.


VMRT

G. Occhetta

Properties



Ir(reducibility)

f ∗ TX ' OP1 (2) ⊕ OP1 (1)⊕d ⊕ OP⊕n−d−1 . 1





2

If X is a prime Fano of dimension n such that rX ≥ (n + 1)/2 then the VMRT at a general point is non degenerate.


VMRT

G. Occhetta

Properties



Ir(reducibility)

f ∗ TX ' OP1 (2) ⊕ OP1 (1)⊕d ⊕ OP⊕n−d−1 . 1





2

If X is a prime Fano of dimension n such that rX ≥ (n + 1)/2 then the VMRT at a general point is non degenerate.

3

The VMRT cannot be an irreducible linear subspace.


VMRT

G. Occhetta

Examples




X

VMRT


VMRT

G. Occhetta

Examples




X Pn

VMRT Pn−1


VMRT

G. Occhetta

Examples




X Pn Qn

VMRT Pn−1 Qn−2


VMRT

G. Occhetta

Examples




X Pn Qn cubic in Pn

VMRT Pn−1 Qn−2 quadric ∩ cubic in Pn−1


VMRT

G. Occhetta

Examples




X Pn Qn cubic in Pn hypersurface Xd ⊂ Pn , d ≤ n

VMRT Pn−1 Qn−2 quadric ∩ cubic in Pn−1 c.i of hypersurfaces of deg 2, . . . , d


VMRT

G. Occhetta

Examples




X Pn Qn cubic in Pn hypersurface Xd ⊂ Pn , d ≤ n G(k, n)

VMRT Pn−1 Qn−2 quadric ∩ cubic in Pn−1 c.i of hypersurfaces of deg 2, . . . , d Pk × Pn−k−1


VMRT

G. Occhetta

Examples




X Pn Qn cubic in Pn hypersurface Xd ⊂ Pn , d ≤ n G(k, n) II G (k, 2m − 1) GII (m − 1, 2m − 1) GIII (k, 2m − 1) III G (m − 1, 2m − 1)

VMRT Pn−1 Qn−2 quadric ∩ cubic in Pn−1 c.i of hypersurfaces of deg 2, . . . , d Pk × Pn−k−1 Pk × Q2m−2k−3 G(1, m − 1) PPm (O(2) ⊕ O(1)2m−2k−2 ) v2 (Pm−1 )


VMRT

G. Occhetta

Examples




X Pn Qn cubic in Pn hypersurface Xd ⊂ Pn , d ≤ n G(k, n) II G (k, 2m − 1) GII (m − 1, 2m − 1) GIII (k, 2m − 1) III G (m − 1, 2m − 1)

VMRT Pn−1 Qn−2 quadric ∩ cubic in Pn−1 c.i of hypersurfaces of deg 2, . . . , d Pk × Pn−k−1 Pk × Q2m−2k−3 G(1, m − 1) PPm (O(2) ⊕ O(1)2m−2k−2 ) v2 (Pm−1 )

GII (m − 1, 2m − 1) orthogonal Grassmannian GIII (k, 2m − 1) symplectic Grassmannian


VMRT Irreducibility




Question (Hwang 2000) Is Cx irreducible if it has positive dimension? Is it at least true for prime Fano manifolds?


VMRT

G. Occhetta

Irreducibility




Question (Hwang 2000) Is Cx irreducible if it has positive dimension? Is it at least true for prime Fano manifolds? • C0 smooth curve with a free Z2 -action.


VMRT

G. Occhetta

Irreducibility




Question (Hwang 2000) Is Cx irreducible if it has positive dimension? Is it at least true for prime Fano manifolds? • C0 smooth curve with a free Z2 -action. • π : C0 → C induced e´ tale covering.


VMRT

G. Occhetta

Irreducibility




Question (Hwang 2000) Is Cx irreducible if it has positive dimension? Is it at least true for prime Fano manifolds? • C0 smooth curve with a free Z2 -action. • π : C0 → C induced e´ tale covering. • Y = Ps × Ps , with a Z2 -action exchanging the factors


VMRT

G. Occhetta

Irreducibility




Question (Hwang 2000) Is Cx irreducible if it has positive dimension? Is it at least true for prime Fano manifolds? • C0 smooth curve with a free Z2 -action. • π : C0 → C induced e´ tale covering. • Y = Ps × Ps , with a Z2 -action exchanging the factors • X 0 = Y × C0 and X quotient by the product action of Z2 .


VMRT

G. Occhetta

Irreducibility





/ C0

X0 π0

X

π

ϕ

/C


VMRT

G. Occhetta

Irreducibility





/ C0

X0 π0

X

π

ϕ

/C

The VRMT at every point of X is Ps−1 t Ps−1 .


VMRT Some results




Theorem (Hwang-Mok 2004) Let X and X 0 be prime Fano manifolds. Assume that at a general point x ∈ X the VMRT Cx is non-linear and of positive dimension.


VMRT Some results




Theorem (Hwang-Mok 2004) Let X and X 0 be prime Fano manifolds. Assume that at a general point x ∈ X the VMRT Cx is non-linear and of positive dimension. Let f : U → U 0 be a biholomorphic map from U ⊂ X onto U 0 ⊂ X 0 such that the differential df sends each irreducible component of C(X)U to an irreducible component of C(X 0 )U 0 biholomorphically.


VMRT Some results




Theorem (Hwang-Mok 2004) Let X and X 0 be prime Fano manifolds. Assume that at a general point x ∈ X the VMRT Cx is non-linear and of positive dimension. Let f : U → U 0 be a biholomorphic map from U ⊂ X onto U 0 ⊂ X 0 such that the differential df sends each irreducible component of C(X)U to an irreducible component of C(X 0 )U 0 biholomorphically. Then f extends to a biholomorphic map F : X → X 0 .


VMRT Some results




Theorem (Hwang-Mok 2004) Let X and X 0 be prime Fano manifolds. Assume that at a general point x ∈ X the VMRT Cx is non-linear and of positive dimension. Let f : U → U 0 be a biholomorphic map from U ⊂ X onto U 0 ⊂ X 0 such that the differential df sends each irreducible component of C(X)U to an irreducible component of C(X 0 )U 0 biholomorphically. Then f extends to a biholomorphic map F : X → X 0 . X has the target rigidity property if, given an holomorphic surjective map f : Y → X, all deformations of f come from automorphisms of the target.


VMRT Some results




Theorem (Hwang-Mok 2004) Let X and X 0 be prime Fano manifolds. Assume that at a general point x ∈ X the VMRT Cx is non-linear and of positive dimension. Let f : U → U 0 be a biholomorphic map from U ⊂ X onto U 0 ⊂ X 0 such that the differential df sends each irreducible component of C(X)U to an irreducible component of C(X 0 )U 0 biholomorphically. Then f extends to a biholomorphic map F : X → X 0 . X has the target rigidity property if, given an holomorphic surjective map f : Y → X, all deformations of f come from automorphisms of the target. Theorem (Hwang-Mok 2004) Let X be a prime Fano manifold, X 6= Pn . Assume that at a general point x ∈ X the VMRT Cx (X) is non-linear and of positive dimension.


VMRT Some results




Theorem (Hwang-Mok 2004) Let X and X 0 be prime Fano manifolds. Assume that at a general point x ∈ X the VMRT Cx is non-linear and of positive dimension. Let f : U → U 0 be a biholomorphic map from U ⊂ X onto U 0 ⊂ X 0 such that the differential df sends each irreducible component of C(X)U to an irreducible component of C(X 0 )U 0 biholomorphically. Then f extends to a biholomorphic map F : X → X 0 . X has the target rigidity property if, given an holomorphic surjective map f : Y → X, all deformations of f come from automorphisms of the target. Theorem (Hwang-Mok 2004) Let X be a prime Fano manifold, X 6= Pn . Assume that at a general point x ∈ X the VMRT Cx (X) is non-linear and of positive dimension. Then X has the target rigidity property.


VMRT Non linearity




Conjecture (Hwang-Mok 2004) Let X be a prime Fano manifold. Then the VMRT of X at a general point is not a union of positive dimensional linear subspaces.




Fano bundles

VMRT and congruences of lines G. Occhetta

Fano bundles




A vector bundle E over a smooth complex projective variety X is a Fano bundle if PX (E) is a Fano manifold.


Fano bundles




A vector bundle E over a smooth complex projective variety X is a Fano bundle if PX (E) is a Fano manifold. If E is a Fano bundle on X then also X is a Fano manifold.


Fano bundles



Congruences



Rank 2 Fano bundles over projective spaces and quadrics have been classified in the 90’s (Ancona, Peternell, Sols, Szurek, Wi´sniewski).


Fano bundles



Congruences



Rank 2 Fano bundles over projective spaces and quadrics have been classified in the 90’s (Ancona, Peternell, Sols, Szurek, Wi´sniewski). Generalization: classify rank 2 Fano bundles over (Fano) manifolds with b2 = b4 = 1. Solved recently by Muñoz, , Solá Conde.


Fano bundles



Congruences



Rank 2 Fano bundles over projective spaces and quadrics have been classified in the 90’s (Ancona, Peternell, Sols, Szurek, Wi´sniewski). Generalization: classify rank 2 Fano bundles over (Fano) manifolds with b2 = b4 = 1. Solved recently by Muñoz, , Solá Conde. As a side effect, a counterexample to the non linearity conjecture has been found.




Fano bundles Contractions


Fano bundles

G. Occhetta

Contractions




Key facts:


Fano bundles

G. Occhetta

Contractions




Key facts: 1

The assumption on b4 provides the following:


Fano bundles

G. Occhetta

Contractions



Key facts: 1



Lemma The second contraction ϕ : P(E) → Y is either


Fano bundles

G. Occhetta

Contractions



Key facts: 1



Lemma The second contraction ϕ : P(E) → Y is either (P) a P1 -bundle;


Fano bundles

G. Occhetta

Contractions



Key facts: 1



Lemma The second contraction ϕ : P(E) → Y is either (P) a P1 -bundle; (C) a conic bundle with reducible fibers, or


Fano bundles

G. Occhetta

Contractions



Key facts: 1



Lemma The second contraction ϕ : P(E) → Y is either (P) a P1 -bundle; (C) a conic bundle with reducible fibers, or (D) the blow-up of a codimension two smooth subvariety.


Fano bundles

G. Occhetta

Contractions



Key facts: 1


Congruences Generalities Special congruences Severi varieties

Lemma The second contraction ϕ : P(E) → Y is either


(P) a P1 -bundle; (C) a conic bundle with reducible fibers, or (D) the blow-up of a codimension two smooth subvariety. 2

Intersection theory combined with number theory provides n = 2, 3 or 5 if ϕ is of fiber type.


Fano bundles A Classification theorem




Theorem Let X be a Fano manifold satisfying b2 = b4 = 1, and let E be an indecomposable rank two Fano bundle on X.






Theorem Let X be a Fano manifold satisfying b2 = b4 = 1, and let E be an indecomposable rank two Fano bundle on X. Then, up to a twist with a line bundle, E is the pull-back of the universal quotient bundle on a Grassmannian G(1, m) by a finite map ψ : X → G(1, m) where either






Theorem Let X be a Fano manifold satisfying b2 = b4 = 1, and let E be an indecomposable rank two Fano bundle on X. Then, up to a twist with a line bundle, E is the pull-back of the universal quotient bundle on a Grassmannian G(1, m) by a finite map ψ : X → G(1, m) where either • ψ is one of the embeddings given by (P1)-(P5) . . . , (D1) P2 ⊂ GII (1, 3)Q3 ⊂ G(1, 4) (lines in Q3 meeting a fixed line), (D2) v2 (P2 ) ⊂ G(1, 3) ((bi)secant lines to v3 (P1 ) ⊂ P3 ), (D3) V53 ⊂ G(1, 4) (trisecant lines to the projection of v2 (P2 ) into P4 ), (C6) . . .


Fano bundles

G. Occhetta

A Classification theorem




Theorem Let X be a Fano manifold satisfying b2 = b4 = 1, and let E be an indecomposable rank two Fano bundle on X. Then, up to a twist with a line bundle, E is the pull-back of the universal quotient bundle on a Grassmannian G(1, m) by a finite map ψ : X → G(1, m) where either • ψ is one of the embeddings given by (P1)-(P5) . . . , (D1) P2 ⊂ GII (1, 3)Q3 ⊂ G(1, 4) (lines in Q3 meeting a fixed line), (D2) v2 (P2 ) ⊂ G(1, 3) ((bi)secant lines to v3 (P1 ) ⊂ P3 ), (D3) V53 ⊂ G(1, 4) (trisecant lines to the projection of v2 (P2 ) into P4 ), (C6) . . . • ψ factorizes by a finite covering ψ1 : X → X1 of one of the

submanifolds of types (P1)-(P5) above and, either (C1) . . . , (C2)-(C5) . . . .


Congruences of lines Generalities




A congruence of lines Pm is subvariety X m−1 ⊂ G(1, m).


Congruences of lines

G. Occhetta

Generalities



Fano bundles Why Fano bundles?

Y := P(Q|X )

A classification theorem

π


X

z

π0

%

Pm



G. Occhetta

Generalities




Y := P(Q|X )


π


X

z

π0

%

Pm


• order of X: number of lines parametrized by X passing through a

general point of Pm , which is zero if π 0 is of fiber type or equal to the degree of π 0 otherwise.



G. Occhetta

Generalities




Y := P(Q|X )


π


X

π0

z

%

Pm



general point of Pm , which is zero if π 0 is of fiber type or equal to the degree of π 0 otherwise. • fundamental point: a point y ∈ Pm such that the fiber π 0−1 (y) has

dimension greater than m − dim(π 0 (Y)).



G. Occhetta

Generalities




Y := P(Q|X )


π


X

π0

z

%

Pm



general point of Pm , which is zero if π 0 is of fiber type or equal to the degree of π 0 otherwise. • fundamental point: a point y ∈ Pm such that the fiber π 0−1 (y) has

dimension greater than m − dim(π 0 (Y)).

• fundamental locus: set of the fundamental points.




Congruences of lines Special congruences



G. Occhetta

Special congruences




Assumptions



G. Occhetta

Special congruences


Assumptions



• X smooth congruence of lines of order 1 in Pm with

Pic(X) ∼ = Z.



G. Occhetta

Special congruences


Assumptions




Pic(X) ∼ = Z.

• Z z ⊂ Pm smooth fundamental locus.



G. Occhetta

Special congruences


Assumptions




Pic(X) ∼ = Z.

• Z z ⊂ Pm smooth fundamental locus. • The images of the fibers of π 0 are linear spaces in G(1, m).



G. Occhetta

Special congruences


Assumptions




Pic(X) ∼ = Z.


Notation



G. Occhetta

Special congruences


Assumptions




Pic(X) ∼ = Z.


Notation • HX generator of Pic(X).



G. Occhetta

Special congruences


Assumptions




Pic(X) ∼ = Z.


Notation • HX generator of Pic(X). • L tautological divisor of P(Q|X ), which is π 0∗ OPm (1).



G. Occhetta

Special congruences


Assumptions




Pic(X) ∼ = Z.


Notation • HX generator of Pic(X). • L tautological divisor of P(Q|X ), which is π 0∗ OPm (1). • H pull-back to Y of the ample generator of Pic(X).



G. Occhetta

Special congruences


Assumptions




Pic(X) ∼ = Z.


Notation • HX generator of Pic(X). • L tautological divisor of P(Q|X ), which is π 0∗ OPm (1). • H pull-back to Y of the ample generator of Pic(X). • E exceptional divisor.



G. Occhetta

Smooth fundamental loci




Proposition



G. Occhetta



Proposition

Ir(reducibility)



• X and Y are Fano manifolds.



G. Occhetta



Proposition

Ir(reducibility)



• X and Y are Fano manifolds. • π 0 is the blow-up of Pm along Z;



G. Occhetta



Proposition

Ir(reducibility)



• X and Y are Fano manifolds. • π 0 is the blow-up of Pm along Z; • X is covered by lines,



G. Occhetta



Proposition

Ir(reducibility)



• X and Y are Fano manifolds. • π 0 is the blow-up of Pm along Z; • X is covered by lines, • HX = OG (1)|X ,



G. Occhetta



Proposition

Ir(reducibility)



• X and Y are Fano manifolds. • π 0 is the blow-up of Pm along Z; • X is covered by lines, • HX = OG (1)|X , • −KX = (m − z)HX .



G. Occhetta



Proposition

Ir(reducibility)



• X and Y are Fano manifolds. • π 0 is the blow-up of Pm along Z; • X is covered by lines, • HX = OG (1)|X , • −KX = (m − z)HX . • f fiber of π : Y → X =⇒ α := E · f = (m − 1)/(m − z − 1).



G. Occhetta



Proposition

Ir(reducibility)



• X and Y are Fano manifolds. • π 0 is the blow-up of Pm along Z; • X is covered by lines, • HX = OG (1)|X , • −KX = (m − z)HX . • f fiber of π : Y → X =⇒ α := E · f = (m − 1)/(m − z − 1). • VMRT of a general x consists of α disjoint linear spaces of

dimension m − z − 2.



G. Occhetta



Proposition

Ir(reducibility)




dimension m − z − 2. Using that E = αL − H and some intersection theory



G. Occhetta



Proposition

Ir(reducibility)




dimension m − z − 2. Using that E = αL − H and some intersection theory • Lk EH m−k+1 = 0 for k > z



G. Occhetta



Proposition

Ir(reducibility)




dimension m − z − 2. Using that E = αL − H and some intersection theory • Lk EH m−k+1 = 0 for k > z • Lz EH m−z+1 = deg Z



G. Occhetta



Proposition

Ir(reducibility)




dimension m − z − 2. Using that E = αL − H and some intersection theory • Lk EH m−k+1 = 0 for k > z • Lz EH m−z+1 = deg Z

we get deg(Z) < αm−z =⇒ Z cannot be a complete intersection.


Congruences of lines Selecting the candidates




Remark The case α = 2 does not happen.






Remark The case α = 2 does not happen. X will be covered by linear spaces of dimension dim X/2, and have index dim X/2 + 1; there are no such Fanos with Pic(X) ' Z.





Congruences



Remark If Hartshorne’s conjecture on complete intersections is true then the possible values of (α, z, m) are:





Congruences



Remark If Hartshorne’s conjecture on complete intersections is true then the possible values of (α, z, m) are: (3, 2k, 3k + 1) with k > 0 , (4, 3, 5), (4, 6, 9) and (5, 4, 6).





Congruences



Remark If Hartshorne’s conjecture on complete intersections is true then the possible values of (α, z, m) are: (3, 2k, 3k + 1) with k > 0 , (4, 3, 5), (4, 6, 9) and (5, 4, 6). Recall: α-secants to a smooth Z of dimension z in Pm .





Congruences



Remark If Hartshorne’s conjecture on complete intersections is true then the possible values of (α, z, m) are: (3, 2k, 3k + 1) with k > 0 , (4, 3, 5), (4, 6, 9) and (5, 4, 6). Recall: α-secants to a smooth Z of dimension z in Pm . The dimension of the VMRT (m − z − 2) of X will be positive in cases





Congruences



Remark If Hartshorne’s conjecture on complete intersections is true then the possible values of (α, z, m) are: (3, 2k, 3k + 1) with k > 0 , (4, 3, 5), (4, 6, 9) and (5, 4, 6). Recall: α-secants to a smooth Z of dimension z in Pm . The dimension of the VMRT (m − z − 2) of X will be positive in cases (3, 2k, 3k + 1) with k > 1 , (4, 6, 9)


Severi varieties




A Severi variety X ⊂ PN is a smooth projective variety of dimension 2 3 (N − 2) which can be isomorphically projected to a projective space of smaller dimension.


Severi varieties





Theorem (Zak) The Severi varieties are


Severi varieties





Theorem (Zak) The Severi varieties are 2 The Veronese surface in P5 ;


Severi varieties





Theorem (Zak) The Severi varieties are 2 The Veronese surface in P5 ; 4 The Segre embedding of P2 × P2 in P8 ;


Severi varieties





Theorem (Zak) The Severi varieties are 2 The Veronese surface in P5 ; 4 The Segre embedding of P2 × P2 in P8 ; 8 The Grassmannian G(1, 5) ⊂ P14 ;


Severi varieties





Theorem (Zak) The Severi varieties are 2 The Veronese surface in P5 ; 4 The Segre embedding of P2 × P2 in P8 ; 8 The Grassmannian G(1, 5) ⊂ P14 ; 16 The variety E16 ⊂ P26 .




Severi varieties And Cremona transformations






Lemma Let S ⊂ P3k+2 be a 2k-dimensional Severi variety.






Lemma Let S ⊂ P3k+2 be a 2k-dimensional Severi variety. The linear system of quadrics containing S provides an involutive Cremona transformation ψ : P3k+2 → P3k+2 , fitting in a diagram:


Severi varieties

G. Occhetta

And Cremona transformations





Severi varieties

B


σ0

σ

| P3k+2

ψ

" / P3k+2


Severi varieties

G. Occhetta

And Cremona transformations





Severi varieties

B


σ0

σ

| P3k+2

ψ

" / P3k+2

where σ and σ 0 denote, respectively, the blowing up of P3k+2 along S and the blowing up of P3k+2 along S0 ∼ = S.


Example Trisecants to projected Severi




Example Let X ⊂ G(1, 3k + 1) be the closure of the family of trisecant lines to a general isomorphic projection Z ⊂ P3k+1 of a 2k-dimensional Severi variety S ⊂ P3k+2 . Then X is a smooth congruence of order one with linear fibers.






For every P ∈ P3k+1 \ Z there exists a unique trisecant line to Z.


Example

G. Occhetta





O


P Z


Example

G. Occhetta





O


O

P

P Z


Example

G. Occhetta





O


O

P

P Z




Example Trisecants to projected Severi Trisecants through P ∈ Z.


Example

G. Occhetta


VMRT Definition

Trisecants through P ∈ Z.

Properties Examples Ir(reducibility)


Congruences

R


O


Q

P Z


Example

G. Occhetta


VMRT Definition




R


O


Q

P Z


Example

G. Occhetta


VMRT Definition




R


O


Q

P Z






X ⊂ G(1, 3k + 1) is a congruence of order one, and Y = PX (Q|X ) is the blow-up of P3k+1 along Z.


Lines in X are the images of the fibers of π 0 : Y → P3k+1 . Lines through a general x ∈ X are the lines contained in three linear spaces passing through x and meeting only in x.






Example

G. Occhetta




Congruences

Y


X

Z


Example

G. Occhetta




Congruences

Y


X

Z


Example

G. Occhetta




Congruences

Y

Z


X

The VMRT at x is the union of three disjoint linear spaces!







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