Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. ·...

Secant Varieties of Segre–Veronese Varieties

Claudiu Raicu

Princeton University

Raleigh, October 2011

Overview

1 Secant Varieties

2 Segre–Veronese Varieties

3 Flattenings

4 Main Results and Techniques

Secant VarietiesDefinitionGiven a subvariety X ⊂ PN , the (k − 1)–st secant variety of X , denotedσk (X ), is the closure of the union of linear subspaces spanned by kpoints on X :

σk (X ) =⋃

x1,··· ,xk∈X

Px1,··· ,xk .

Alternatively, write PN = PW for some vector space W , and let X ⊂Wdenote the cone over X . The cone σk (X ) over σk (X ) is the closure ofthe image of the map

s : X × · · · × X −→W ,

s(x1, · · · , xk ) = x1 + · · ·+ xk .

ProblemGiven (the equations of) X , determine (the equations of) σk (X ).

σk (X ) =⋃

x1,··· ,xk∈X

Px1,··· ,xk .

s : X × · · · × X −→W ,

s(x1, · · · , xk ) = x1 + · · ·+ xk .

σk (X ) =⋃

x1,··· ,xk∈X

Px1,··· ,xk .

s : X × · · · × X −→W ,

s(x1, · · · , xk ) = x1 + · · ·+ xk .

Solution to Problem

The morphism s of affine varieties corresponds to a ring map

s# : Sym(W ∗)→ K [X × · · · × X ] = K [X ]⊗ · · · ⊗ K [X ].

I(σk (X )) and K [σk (X )] are the kernel and image respectively of s#.

Big Problem

Computing the kernel and image of s# is really hard!

Interesting examples:

1 curves;2 toric varieties;3 homogeneous spaces;4 Grassmannians;5 Segre and Veronese varieties.

Solution to Problem

s# : Sym(W ∗)→ K [X × · · · × X ] = K [X ]⊗ · · · ⊗ K [X ].

Big Problem

Solution to Problem

s# : Sym(W ∗)→ K [X × · · · × X ] = K [X ]⊗ · · · ⊗ K [X ].

Big Problem

1 curves;

2 toric varieties;3 homogeneous spaces;4 Grassmannians;5 Segre and Veronese varieties.

Solution to Problem

s# : Sym(W ∗)→ K [X × · · · × X ] = K [X ]⊗ · · · ⊗ K [X ].

Big Problem

1 curves;2 toric varieties;

3 homogeneous spaces;4 Grassmannians;5 Segre and Veronese varieties.

Solution to Problem

s# : Sym(W ∗)→ K [X × · · · × X ] = K [X ]⊗ · · · ⊗ K [X ].

Big Problem

1 curves;2 toric varieties;3 homogeneous spaces;

4 Grassmannians;5 Segre and Veronese varieties.

Solution to Problem

s# : Sym(W ∗)→ K [X × · · · × X ] = K [X ]⊗ · · · ⊗ K [X ].

Big Problem

1 curves;2 toric varieties;3 homogeneous spaces;4 Grassmannians;

5 Segre and Veronese varieties.

Solution to Problem

s# : Sym(W ∗)→ K [X × · · · × X ] = K [X ]⊗ · · · ⊗ K [X ].

Big Problem

Segre–Veronese VarietiesConsider vector spaces Vi , i = 1, · · · ,n with duals V ∗i , and positiveintegers d1, · · · ,dn. We let

X = PV ∗1 × · · · × PV ∗n

and think of it as a subvariety in projective space via the embeddingdetermined by the line bundle OX (d1, · · · ,dn).

X is the image of

SVd1,··· ,dn : PV ∗1 × · · · × PV ∗n → P(Symd1 V ∗1 ⊗ · · · ⊗ Symdn V ∗n ),

([e1], · · · , [en]) 7→ [ed11 ⊗ · · · ⊗ edn

We call X a Segre–Veronese variety. Write W ∗ for the linear forms onthe target of SVd1,··· ,dn , W ∗ = Symd1 V1 ⊗ · · · ⊗ Symdn Vn. To computethe equations of σk (X ) it’s “enough” to understand the kernel of

s# : Sym(W ∗) −→

⊕r≥0

Symrd1 V1 ⊗ · · · ⊗ Symrdn Vn

X = PV ∗1 × · · · × PV ∗n

and think of it as a subvariety in projective space via the embeddingdetermined by the line bundle OX (d1, · · · ,dn). X is the image of

([e1], · · · , [en]) 7→ [ed11 ⊗ · · · ⊗ edn

We call X a Segre–Veronese variety.

Write W ∗ for the linear forms onthe target of SVd1,··· ,dn , W ∗ = Symd1 V1 ⊗ · · · ⊗ Symdn Vn. To computethe equations of σk (X ) it’s “enough” to understand the kernel of

s# : Sym(W ∗) −→

⊕r≥0

X = PV ∗1 × · · · × PV ∗n

and think of it as a subvariety in projective space via the embeddingdetermined by the line bundle OX (d1, · · · ,dn). X is the image of

([e1], · · · , [en]) 7→ [ed11 ⊗ · · · ⊗ edn

We call X a Segre–Veronese variety. Write W ∗ for the linear forms onthe target of SVd1,··· ,dn , W ∗ = Symd1 V1 ⊗ · · · ⊗ Symdn Vn. To computethe equations of σk (X ) it’s “enough” to understand the kernel of

s# : Sym(W ∗) −→

⊕r≥0

Example: generic matrices, flatteningsWhen all di = 1, X is the Segre variety (pure tensors). When n = 2 weget matrices of rank 1 as the image of

SV1,1 : PV ∗1 × PV ∗2 → P(V ∗1 ⊗ V ∗2 ).

More generally, σk (X ) is the collection of matrices of rank at most k ,which are defined by the vanishing of their (k + 1)–minors.If n = 3, dim(Vi) = 3, the ambient space consists of 3× 3× 3 tensorsT = (xijk ), which we can flatten by thinking of V ∗1 ⊗ V ∗2 as a singlefactor:

V ∗1 ⊗ V ∗2 ⊗ V ∗3 = (V ∗1 ⊗ V ∗2 )⊗ V ∗3 .

The tensor T flattens to a 3× 9 matrix x11,1 x12,1 x13,1 x21,1 x22,1 x23,1 x31,1 x32,1 x33,1x11,2 x12,2 x13,2 x21,2 x22,2 x23,2 x31,2 x32,2 x33,2x11,3 x12,3 x13,3 x21,3 x22,3 x23,3 x31,3 x32,3 x33,3

SV1,1 : PV ∗1 × PV ∗2 → P(V ∗1 ⊗ V ∗2 ).

More generally, σk (X ) is the collection of matrices of rank at most k ,which are defined by the vanishing of their (k + 1)–minors.

If n = 3, dim(Vi) = 3, the ambient space consists of 3× 3× 3 tensorsT = (xijk ), which we can flatten by thinking of V ∗1 ⊗ V ∗2 as a singlefactor:

V ∗1 ⊗ V ∗2 ⊗ V ∗3 = (V ∗1 ⊗ V ∗2 )⊗ V ∗3 .

SV1,1 : PV ∗1 × PV ∗2 → P(V ∗1 ⊗ V ∗2 ).

V ∗1 ⊗ V ∗2 ⊗ V ∗3 = (V ∗1 ⊗ V ∗2 )⊗ V ∗3 .

SV1,1 : PV ∗1 × PV ∗2 → P(V ∗1 ⊗ V ∗2 ).

V ∗1 ⊗ V ∗2 ⊗ V ∗3 = (V ∗1 ⊗ V ∗2 )⊗ V ∗3 .

A conjecture about flattenings1 σ1(P2 × P2 × P2): 2–minors of flattenings. (Kostant)

2 σ2(P2 × P2 × P2): 3–minors of flattenings. (Landsberg–Manivel)3 σ3(P2 × P2 × P2): 4–minors of flattenings give nothing. Instead,

use Strassen’s commutation conditions. (Landsberg–Weyman)

Conjecture (Garcia–Stillman–Sturmfels, Pachter–Sturmfels)The 3–minors of flattenings generate the ideal of σ2(X ) when X is aSegre variety.

Known cases:

1 2 factors (classical)2 3 factors, and the set–theoretic version for any number of factors

(Landsberg and Manivel)3 4 factors (Landsberg and Weyman)4 5 factors (Allman and Rhodes)

A conjecture about flattenings1 σ1(P2 × P2 × P2): 2–minors of flattenings. (Kostant)2 σ2(P2 × P2 × P2): 3–minors of flattenings. (Landsberg–Manivel)

3 σ3(P2 × P2 × P2): 4–minors of flattenings give nothing. Instead,use Strassen’s commutation conditions. (Landsberg–Weyman)

Known cases:

A conjecture about flattenings1 σ1(P2 × P2 × P2): 2–minors of flattenings. (Kostant)2 σ2(P2 × P2 × P2): 3–minors of flattenings. (Landsberg–Manivel)3 σ3(P2 × P2 × P2): 4–minors of flattenings

give nothing. Instead,use Strassen’s commutation conditions. (Landsberg–Weyman)

Known cases:

A conjecture about flattenings1 σ1(P2 × P2 × P2): 2–minors of flattenings. (Kostant)2 σ2(P2 × P2 × P2): 3–minors of flattenings. (Landsberg–Manivel)3 σ3(P2 × P2 × P2): 4–minors of flattenings give nothing. Instead,

Known cases:

1 2 factors (classical)

2 3 factors, and the set–theoretic version for any number of factors(Landsberg and Manivel)

3 4 factors (Landsberg and Weyman)4 5 factors (Allman and Rhodes)

Known cases:

(Landsberg and Manivel)

3 4 factors (Landsberg and Weyman)4 5 factors (Allman and Rhodes)

Known cases:

(Landsberg and Manivel)3 4 factors (Landsberg and Weyman)

4 5 factors (Allman and Rhodes)

Known cases:

Example: Veronese embeddings of P1

When n = 1, write V = V1, d = d1. If dim(V ) = 2 (with basis {x , y} ofV ∗), X is a rational normal curve of degree d , embedded by

[x : y ] −→ [xd : xd−1 · y : · · · : x · yd−1 : yd ].

Write zi ∈ Symd V for the coordinate function of the ambient projectivespace P(Symd V ∗) corresponding to xd−i · y i .

We obtain symmetricflattenings (or catalecticant matrices Cat(a,b)) by writing themultiplication table of Syma V ⊗ Symb V → Symd V . For d = 6, we get

Cat(3,3) :

x3 x2 · y x · y2 y3

x3 z0 z1 z2 z3x2 · y z1 z2 z3 z4x · y2 z2 z3 z4 z5

y3 z3 z4 z5 z6

Cat(2,4) :

z0 z1 z2 z3 z4z1 z2 z3 z4 z5z2 z3 z4 z5 z6

Cat(5,1) :

z0 z1z1 z2z2 z3z3 z4z4 z5z5 z6

[x : y ] −→ [xd : xd−1 · y : · · · : x · yd−1 : yd ].

Write zi ∈ Symd V for the coordinate function of the ambient projectivespace P(Symd V ∗) corresponding to xd−i · y i . We obtain symmetricflattenings (or catalecticant matrices Cat(a,b)) by writing themultiplication table of Syma V ⊗ Symb V → Symd V .

For d = 6, we get

Cat(3,3) :

x3 x2 · y x · y2 y3

y3 z3 z4 z5 z6

Cat(2,4) :

Cat(5,1) :

[x : y ] −→ [xd : xd−1 · y : · · · : x · yd−1 : yd ].

Write zi ∈ Symd V for the coordinate function of the ambient projectivespace P(Symd V ∗) corresponding to xd−i · y i . We obtain symmetricflattenings (or catalecticant matrices Cat(a,b)) by writing themultiplication table of Syma V ⊗ Symb V → Symd V . For d = 6, we get

Cat(3,3) :

x3 x2 · y x · y2 y3

y3 z3 z4 z5 z6

Cat(2,4) :

Cat(5,1) :

[x : y ] −→ [xd : xd−1 · y : · · · : x · yd−1 : yd ].

Cat(3,3) :

x3 x2 · y x · y2 y3

y3 z3 z4 z5 z6

Cat(2,4) :

Cat(5,1) :

[x : y ] −→ [xd : xd−1 · y : · · · : x · yd−1 : yd ].

Cat(3,3) :

x3 x2 · y x · y2 y3

y3 z3 z4 z5 z6

Cat(2,4) :

Cat(5,1) :

Veronese varieties

Theorem (Gruson–Peskine, Eisenbud, Conca)If X is a rational normal curve of degree d, then I(σk (X )) is generatedby the (k + 1)–minors of any Cat(a,b), where a,b ≥ k, a + b = d.

Now assume that dim(V ) is arbitrary. We can still talk aboutcatalecticant matrices Cat(a,b) whenever a + b = d .

1 (k + 1)–minors of catalecticants vanish on σk (X ).2 X = σ1(X ) is defined by the 2–minors of any Cat(a,b). (Pucci)3 σ2(X ) is defined by the 3–minors of Cat(1,d − 1) and

Cat(2,d − 2). (Kanev)4 σk (X ) is NOT defined by (k + 1)–minors of catalecticants in

general. (Buczynska–Buczynski)

Conjecture (Geramita)The ideals of 3–minors of Cat(a,b) are all equal for a,b ≥ 2.

Veronese varieties

1 (k + 1)–minors of catalecticants vanish on σk (X ).

2 X = σ1(X ) is defined by the 2–minors of any Cat(a,b). (Pucci)3 σ2(X ) is defined by the 3–minors of Cat(1,d − 1) and

Veronese varieties

1 (k + 1)–minors of catalecticants vanish on σk (X ).2 X = σ1(X ) is defined by the 2–minors of any Cat(a,b). (Pucci)

3 σ2(X ) is defined by the 3–minors of Cat(1,d − 1) andCat(2,d − 2). (Kanev)

4 σk (X ) is NOT defined by (k + 1)–minors of catalecticants ingeneral. (Buczynska–Buczynski)

Veronese varieties

Cat(2,d − 2). (Kanev)

4 σk (X ) is NOT defined by (k + 1)–minors of catalecticants ingeneral. (Buczynska–Buczynski)

Veronese varieties

Main Results

Theorem (–)Geramita conjecture holds, as well as its generalization to 4–minors.

QuestionAre the ideals of k–minors of Cat(a,b) all equal for a,b ≥ k − 1?

Theorem (–)For X a Segre–Veronese variety, the ideal of σ2(X ) is generated by3–minors of flattenings. Moreover, one has an explicit description ofthe multiplicities of the irreducible representations that occur in thedecomposition of the homogeneous coordinate ring of σ2(X ).

Main Results

Polarization and Specialization

Suppose n = 2, d1 = 2, d2 = 1, and focus on the equations of degree4 of σ2(X ). We look for the kernel of

s# : Sym4(Sym2 V1 ⊗ V2) −→⊕a+b=4

(Sym2a V1 ⊗ Syma V2)⊗ (Sym2b V1 ⊗ Symb V2).

“Representation theory yoga” =⇒ free to choose mi = dim(Vi)arbitrarily, as long as mi ≥ 2. Take m1 = 8, m2 = 4. The (SL-)zero–weight spaces S and T of the source and target of s# arerepresentations of the Weyl group S8 × S4. Enough to analyze

s#0 : S −→ T .

To do that, use the representation theory of (products of) symmetricgroups, and the combinatorics that comes with it.

s# : Sym4(Sym2 V1 ⊗ V2) −→⊕a+b=4

“Representation theory yoga” =⇒ free to choose mi = dim(Vi)arbitrarily, as long as mi ≥ 2.

Take m1 = 8, m2 = 4. The (SL-)zero–weight spaces S and T of the source and target of s# arerepresentations of the Weyl group S8 × S4. Enough to analyze

s#0 : S −→ T .

s# : Sym4(Sym2 V1 ⊗ V2) −→⊕a+b=4

“Representation theory yoga” =⇒ free to choose mi = dim(Vi)arbitrarily, as long as mi ≥ 2. Take m1 = 8, m2 = 4.

The (SL-)zero–weight spaces S and T of the source and target of s# arerepresentations of the Weyl group S8 × S4. Enough to analyze

s#0 : S −→ T .

s# : Sym4(Sym2 V1 ⊗ V2) −→⊕a+b=4

s#0 : S −→ T .

s# : Sym4(Sym2 V1 ⊗ V2) −→⊕a+b=4

s#0 : S −→ T .

Polarization and SpecializationA typical monomial in S looks like

(x1x2 ⊗ y2) · (x3x6 ⊗ y1) · (x4x7 ⊗ y4) · (x5x8 ⊗ y3),

((xi)i and (yj)j are bases for V1,V2).

It specializes to

m = (x21 ⊗ y2) · (x3x2 ⊗ y1) · (x2x3 ⊗ y2) · (x2

3 ⊗ y2)

via the specialization map φ that sends

{x1, x2} → x1, {x4, x6} → x2, {x3, x5, x7, x8} → x3,

{y1} → y1, {y2, y3, y4} → y2.

Any kernel element of s#0 specializes to a kernel element of s#. We

can polarize m by

m 7→ average(m0 : φ(m0) = m).

Any kernel element of s# polarizes to a kernel element of s#0 .

(x1x2 ⊗ y2) · (x3x6 ⊗ y1) · (x4x7 ⊗ y4) · (x5x8 ⊗ y3),

((xi)i and (yj)j are bases for V1,V2). It specializes to

m = (x21 ⊗ y2) · (x3x2 ⊗ y1) · (x2x3 ⊗ y2) · (x2

3 ⊗ y2)

{x1, x2} → x1, {x4, x6} → x2, {x3, x5, x7, x8} → x3,

{y1} → y1, {y2, y3, y4} → y2.

can polarize m by

(x1x2 ⊗ y2) · (x3x6 ⊗ y1) · (x4x7 ⊗ y4) · (x5x8 ⊗ y3),

m = (x21 ⊗ y2) · (x3x2 ⊗ y1) · (x2x3 ⊗ y2) · (x2

3 ⊗ y2)

{x1, x2} → x1, {x4, x6} → x2, {x3, x5, x7, x8} → x3,

{y1} → y1, {y2, y3, y4} → y2.

Any kernel element of s#0 specializes to a kernel element of s#.

Wecan polarize m by

(x1x2 ⊗ y2) · (x3x6 ⊗ y1) · (x4x7 ⊗ y4) · (x5x8 ⊗ y3),

m = (x21 ⊗ y2) · (x3x2 ⊗ y1) · (x2x3 ⊗ y2) · (x2

3 ⊗ y2)

{x1, x2} → x1, {x4, x6} → x2, {x3, x5, x7, x8} → x3,

{y1} → y1, {y2, y3, y4} → y2.

can polarize m by

(x1x2 ⊗ y2) · (x3x6 ⊗ y1) · (x4x7 ⊗ y4) · (x5x8 ⊗ y3),

m = (x21 ⊗ y2) · (x3x2 ⊗ y1) · (x2x3 ⊗ y2) · (x2

3 ⊗ y2)

{x1, x2} → x1, {x4, x6} → x2, {x3, x5, x7, x8} → x3,

{y1} → y1, {y2, y3, y4} → y2.

can polarize m by

Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. ·...

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