secant varieties of segre veronese varietiescraicu/slides/secantssv.pdf · 2014. 9. 19. ·...
TRANSCRIPT
![Page 1: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/1.jpg)
Secant Varieties of Segre–Veronese Varieties
Claudiu Raicu
Princeton University
Raleigh, October 2011
![Page 2: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/2.jpg)
Overview
1 Secant Varieties
2 Segre–Veronese Varieties
3 Flattenings
4 Main Results and Techniques
![Page 3: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/3.jpg)
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 ).
![Page 4: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/4.jpg)
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 ).
![Page 5: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/5.jpg)
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 ).
![Page 6: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/6.jpg)
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.
![Page 7: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/7.jpg)
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.
![Page 8: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/8.jpg)
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.
![Page 9: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/9.jpg)
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.
![Page 10: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/10.jpg)
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.
![Page 11: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/11.jpg)
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.
![Page 12: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/12.jpg)
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.
![Page 13: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/13.jpg)
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
n ].
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
⊗k
.
![Page 14: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/14.jpg)
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
n ].
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
⊗k
.
![Page 15: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/15.jpg)
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
n ].
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
⊗k
.
![Page 16: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/16.jpg)
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
![Page 17: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/17.jpg)
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
![Page 18: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/18.jpg)
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
![Page 19: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/19.jpg)
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
![Page 20: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/20.jpg)
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)
![Page 21: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/21.jpg)
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)
![Page 22: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/22.jpg)
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)
![Page 23: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/23.jpg)
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)
![Page 24: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/24.jpg)
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)
![Page 25: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/25.jpg)
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)
![Page 26: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/26.jpg)
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)
![Page 27: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/27.jpg)
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)
![Page 28: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/28.jpg)
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)
![Page 29: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/29.jpg)
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
![Page 30: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/30.jpg)
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
![Page 31: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/31.jpg)
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
![Page 32: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/32.jpg)
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
![Page 33: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/33.jpg)
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
![Page 34: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/34.jpg)
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.
![Page 35: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/35.jpg)
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.
![Page 36: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/36.jpg)
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.
![Page 37: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/37.jpg)
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) andCat(2,d − 2). (Kanev)
4 σk (X ) is NOT defined by (k + 1)–minors of catalecticants ingeneral. (Buczynska–Buczynski)
Conjecture (Geramita)The ideals of 3–minors of Cat(a,b) are all equal for a,b ≥ 2.
![Page 38: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/38.jpg)
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 ingeneral. (Buczynska–Buczynski)
Conjecture (Geramita)The ideals of 3–minors of Cat(a,b) are all equal for a,b ≥ 2.
![Page 39: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/39.jpg)
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.
![Page 40: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/40.jpg)
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.
![Page 41: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/41.jpg)
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 ).
![Page 42: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/42.jpg)
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 ).
![Page 43: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/43.jpg)
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 ).
![Page 44: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/44.jpg)
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.
![Page 45: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/45.jpg)
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.
![Page 46: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/46.jpg)
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.
![Page 47: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/47.jpg)
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.
![Page 48: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/48.jpg)
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.
![Page 49: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/49.jpg)
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 .
![Page 50: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/50.jpg)
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 .
![Page 51: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/51.jpg)
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#.
Wecan polarize m by
m 7→ average(m0 : φ(m0) = m).
Any kernel element of s# polarizes to a kernel element of s#0 .
![Page 52: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/52.jpg)
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 .
![Page 53: Secant Varieties of Segre Veronese Varietiescraicu/slides/secantsSV.pdf · 2014. 9. 19. · Segre–Veronese Varieties Consider vector spaces Vi, i = 1; ;n with duals V i, and positive](https://reader036.vdocuments.mx/reader036/viewer/2022070212/61082a29743dc866e647e98a/html5/thumbnails/53.jpg)
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 .