cactus varieties of cubic forms: apolar local artinian ...€¦ · cactus varieties of cubic forms:...
TRANSCRIPT
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Cactus Varieties of Cubic Forms:
Apolar Local Artinian Gorenstein Rings
[–, J. Jelisiejew, P. Macias Marques, K. Ranestad]
Alessandra Bernardi
Univeristy of Bologna (Italy)
November 13th, 2014
Tensors in Computer Science and Geometry
Simons Institute, Berkeley1 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Brief
1 Waring rank and Cactus rank
2 Theorem: Dimensions of Cactus varieties for cubic forms
3 Apolarity
4 Local apolar schemes
5 Proof of the Theorem
2 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Waring rank
F ∈ Sd = K [x0, . . . , xn]d homog. of deg d (charK 6= 2, 3)
Definition
Waring rank: min r ∈ N s.t. F = Ld1 + · · ·+ Ld
r with Li ∈ S1
Veronese: νd : P(S1)→ P(Sd), [L] 7→ [Ld ]
min r ∈ N s.t. [F ] ∈ r -th secant space to νd(P(S1))
The shortest length of a smooth finite scheme Γ ⊂ P(S1) s.t.
[F ] ∈ 〈νd(Γ)〉, Γ = {[L1], . . . , [Lr ]}
3 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Cactus rank
Remove “smooth”
Definition
Cactus Rank: min r ∈ N s.t. ∃ finite (Gor.) scheme Γ of length r
s.t. [F ] ∈ 〈νd(Γ)〉
Secr (νd(P(S1))) = ∪Γ∈HilbrP(S1),Γsmooth〈νd(Γ)〉 Secant variety to
Veronese
Cactusr (νd(P(S1))) = ∪Γ∈HilbrP(S1)〈νd(Γ)〉 Cactus variety
4 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Cactus rank
Remove “smooth”
Definition
Cactus Rank: min r ∈ N s.t. ∃ finite (Gor.) scheme Γ of length r
s.t. [F ] ∈ 〈νd(Γ)〉
Secr (νd(P(S1))) = ∪Γ∈HilbrP(S1),Γsmooth〈νd(Γ)〉 Secant variety to
Veronese
Cactusr (νd(P(S1))) = ∪Γ∈HilbrP(S1)〈νd(Γ)〉 Cactus variety
4 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Theorem
We focus on CUBIC forms
Theorem
When r ≤ 17, then
dimCactusr (X3,n) = dimSecr (X3,n).
When 18 ≤ r ≤ 2n + 2, then dimCactusr (X3,n) > dim Secr (X3,n) and
dimCactusr (X3,n) =
=
min{
148
r 3 − 38r 2 + rn + 5
3r − 2,
(n+3
3
)− 1}, if r ≥ 18 even,
min{
148
r 3 − 716
r 2 + rn + 11948
r − 6516,(n+3
3
)− 1}, if r ≥ 18 odd.
[CNJ] : r ≤ 13 Any local scheme is smoothable (=flat limit of smooth)
5 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Theorem
We focus on CUBIC forms
Theorem
When r ≤ 17, then
dimCactusr (X3,n) = dimSecr (X3,n).
When 18 ≤ r ≤ 2n + 2, then dimCactusr (X3,n) > dim Secr (X3,n) and
dimCactusr (X3,n) =
=
min{
148
r 3 − 38r 2 + rn + 5
3r − 2,
(n+3
3
)− 1}, if r ≥ 18 even,
min{
148
r 3 − 716
r 2 + rn + 11948
r − 6516,(n+3
3
)− 1}, if r ≥ 18 odd.
[CNJ] : r ≤ 13 Any local scheme is smoothable (=flat limit of smooth)
5 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Apolarity
S = K [x0, . . . , xn], T = K [y0, . . . , yn]
yα(x [β])
=
x [β−α] if β ≥ α,
0 otherwise.
S1 and T1 are dual spaces.
T is naturally the coordinate ring of P(S1).
Definition
f ∈ S . Apolar ideal: f ⊥ = {ϕ ∈ T |ϕ(f ) = 0}.
Γ ⊂ P(S1) is apolar to F ∈ S if IΓ ⊂ F⊥ ⊂ T .
Lemma (Apolarity Lemma)
Γ ⊂ P(S1) is apolar to F ∈ Sd iff [F ] ∈ 〈νd(Γ)〉 ⊂ P(Sd).
6 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Apolarity
S = K [x0, . . . , xn], T = K [y0, . . . , yn]
yα(x [β])
=
x [β−α] if β ≥ α,
0 otherwise.
S1 and T1 are dual spaces.
T is naturally the coordinate ring of P(S1).
Definition
f ∈ S . Apolar ideal: f ⊥ = {ϕ ∈ T |ϕ(f ) = 0}.
Γ ⊂ P(S1) is apolar to F ∈ S if IΓ ⊂ F⊥ ⊂ T .
Lemma (Apolarity Lemma)
Γ ⊂ P(S1) is apolar to F ∈ Sd iff [F ] ∈ 〈νd(Γ)〉 ⊂ P(Sd).
6 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Apolarity
S = K [x0, . . . , xn], T = K [y0, . . . , yn]
yα(x [β])
=
x [β−α] if β ≥ α,
0 otherwise.
S1 and T1 are dual spaces.
T is naturally the coordinate ring of P(S1).
Definition
f ∈ S . Apolar ideal: f ⊥ = {ϕ ∈ T |ϕ(f ) = 0}.
Γ ⊂ P(S1) is apolar to F ∈ S if IΓ ⊂ F⊥ ⊂ T .
Lemma (Apolarity Lemma)
Γ ⊂ P(S1) is apolar to F ∈ Sd iff [F ] ∈ 〈νd(Γ)〉 ⊂ P(Sd).
6 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Apolarity
S = K [x0, . . . , xn], T = K [y0, . . . , yn]
yα(x [β])
=
x [β−α] if β ≥ α,
0 otherwise.
S1 and T1 are dual spaces.
T is naturally the coordinate ring of P(S1).
Definition
f ∈ S . Apolar ideal: f ⊥ = {ϕ ∈ T |ϕ(f ) = 0}.
Γ ⊂ P(S1) is apolar to F ∈ S if IΓ ⊂ F⊥ ⊂ T .
Lemma (Apolarity Lemma)
Γ ⊂ P(S1) is apolar to F ∈ Sd iff [F ] ∈ 〈νd(Γ)〉 ⊂ P(Sd).
6 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Apolarity
crk(F ) = min r ∈ N s.t. ∃ Γ ∈ Hilbr (P(S1)) : [F ] ∈ 〈νd(Γ)〉
crk(F ) = min r ∈ N s.t. ∃ Γ ∈ Hilbr (P(S1)) : Γ is apolar to F
crk(F ) = min r ∈ N s.t. ∃ Γ ∈ Hilbr (P(S1)) : IΓ ⊂ F⊥
7 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Local apolar scheme
Properties: Tf := T/f ⊥ local Artinian Gorenstein ring ([IK]):
Local: The image of T1 in Tf generates the only max ideal m;
Artinian: Tf is finitely generated as K -mod;
Gor: Tf has 1-dim’l socle (the annihilator of the max ideal).
If F is homog. ⇒ T/F⊥ graded.
8 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Local apolar scheme
F ∈ Sd ⇒ Γ apolar scheme locally Gor. ⇒
Γ = Γ1 ∪ · · · ∪ Γs , with Γi local A.G.
F = F1 + · · ·+ Fs s.t. Γi apolar to Fi
More generally: any Local AG scheme Γi is the AFFINE apolar
scheme of a poly gi ∈ S (unique up to a unit in the ring of diff.
operators): for any LAG T 0/I ,∃ g ∈ S0s.t.I = g⊥.9 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Local apolar scheme
F ∈ Sd ⇒ Γ apolar scheme locally Gor. ⇒
Γ = Γ1 ∪ · · · ∪ Γs , with Γi local A.G.
F = F1 + · · ·+ Fs s.t. Γi apolar to Fi
More generally: any Local AG scheme Γi is the AFFINE apolar
scheme of a poly gi ∈ S (unique up to a unit in the ring of diff.
operators): for any LAG T 0/I ,∃ g ∈ S0s.t.I = g⊥.9 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
F defines Γ:
F ∈ Sd ⇒ Γ = Γ1 ∪ · · · ∪ Γs , with Γi local A.G.
Γ is defined by gi ’s:
Γ = Γ1 ∪ · · · ∪ Γs ⇐ g1 + · · ·+ gs s.t. Γi AFFINE apolar to gi
Plan of the proof of the Theorem:
1 Understand the link between F and the gi ’s.
10 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
F defines Γ:
F ∈ Sd ⇒ Γ = Γ1 ∪ · · · ∪ Γs , with Γi local A.G.
Γ is defined by gi ’s:
Γ = Γ1 ∪ · · · ∪ Γs ⇐ g1 + · · ·+ gs s.t. Γi AFFINE apolar to gi
Plan of the proof of the Theorem:
1 Understand the link between F and the gi ’s.
10 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
1 Understand the link between F and the gi ’s.
g = g (0) + · · ·+ g (d)︸ ︷︷ ︸deg d tail of g
+gd+1 + · · ·+ g (l), deg(g (i)) = i .
Proposition
F ∈ Sd , f = F (1, x1, . . . , xn). Let Γ be a scheme of minimal length
among local schemes supported at [l ] = [1 : 0 : . . . : 0] that are
apolar to F ⇒ Γ is the AFFINE apolar scheme to a poly
g ∈ K [x1, . . . , xn] whose deg d tail equals f .
So g may be chosen s.t. f is its tail (: Γ is also defined by many g ’s
that does not have f as a tail)11 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
1 Understand the link between F and the gi ’s.
g = g (0) + · · ·+ g (d)︸ ︷︷ ︸deg d tail of g
+gd+1 + · · ·+ g (l), deg(g (i)) = i .
Proposition
F ∈ Sd , f = F (1, x1, . . . , xn). Let Γ be a scheme of minimal length
among local schemes supported at [l ] = [1 : 0 : . . . : 0] that are
apolar to F ⇒ Γ is the AFFINE apolar scheme to a poly
g ∈ K [x1, . . . , xn] whose deg d tail equals f .
So g may be chosen s.t. f is its tail (: Γ is also defined by many g ’s
that does not have f as a tail)11 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Proof of Proposition
Natural apolar scheme of F at [l ]: ZF ,l := V (f ⊥)
Lemma (Buczynski)
Z local supp at [l ] apolar to F ⇒ ∃Z ′ ⊂ Z apolar to F s.t. Z ′ = ZG ,l
for some G ∈ S . Moreover F = Ψ(G ) for some Ψ ∈ T .
⇒ we can choose Γ = ZG ,l for some G s.t. F = Ψ(G ).
If F = Ψ(G ) then f and ψ(g) have the same deg d tail.
Diff (ψ(g)) ⊂ Diff (g)⇒ Zψ(g) ⊂ Γ.
Γ minimal ⇒ suff. to see that Zψ(g) apolar F
(ψ′(ψ(g)) = 0⇒ Ψ′(Ψ(G )) = 0.)
212 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Proof of Proposition
Natural apolar scheme of F at [l ]: ZF ,l := V (f ⊥)
Lemma (Buczynski)
Z local supp at [l ] apolar to F ⇒ ∃Z ′ ⊂ Z apolar to F s.t. Z ′ = ZG ,l
for some G ∈ S . Moreover F = Ψ(G ) for some Ψ ∈ T .
⇒ we can choose Γ = ZG ,l for some G s.t. F = Ψ(G ).
If F = Ψ(G ) then f and ψ(g) have the same deg d tail.
Diff (ψ(g)) ⊂ Diff (g)⇒ Zψ(g) ⊂ Γ.
Γ minimal ⇒ suff. to see that Zψ(g) apolar F
(ψ′(ψ(g)) = 0⇒ Ψ′(Ψ(G )) = 0.)
212 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Proof of Proposition
Natural apolar scheme of F at [l ]: ZF ,l := V (f ⊥)
Lemma (Buczynski)
Z local supp at [l ] apolar to F ⇒ ∃Z ′ ⊂ Z apolar to F s.t. Z ′ = ZG ,l
for some G ∈ S . Moreover F = Ψ(G ) for some Ψ ∈ T .
⇒ we can choose Γ = ZG ,l for some G s.t. F = Ψ(G ).
If F = Ψ(G ) then f and ψ(g) have the same deg d tail.
Diff (ψ(g)) ⊂ Diff (g)⇒ Zψ(g) ⊂ Γ.
Γ minimal ⇒ suff. to see that Zψ(g) apolar F
(ψ′(ψ(g)) = 0⇒ Ψ′(Ψ(G )) = 0.)
212 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Proof of Proposition
Natural apolar scheme of F at [l ]: ZF ,l := V (f ⊥)
Lemma (Buczynski)
Z local supp at [l ] apolar to F ⇒ ∃Z ′ ⊂ Z apolar to F s.t. Z ′ = ZG ,l
for some G ∈ S . Moreover F = Ψ(G ) for some Ψ ∈ T .
⇒ we can choose Γ = ZG ,l for some G s.t. F = Ψ(G ).
If F = Ψ(G ) then f and ψ(g) have the same deg d tail.
Diff (ψ(g)) ⊂ Diff (g)⇒ Zψ(g) ⊂ Γ.
Γ minimal ⇒ suff. to see that Zψ(g) apolar F
(ψ′(ψ(g)) = 0⇒ Ψ′(Ψ(G )) = 0.)
212 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Proof of Proposition
Natural apolar scheme of F at [l ]: ZF ,l := V (f ⊥)
Lemma (Buczynski)
Z local supp at [l ] apolar to F ⇒ ∃Z ′ ⊂ Z apolar to F s.t. Z ′ = ZG ,l
for some G ∈ S . Moreover F = Ψ(G ) for some Ψ ∈ T .
⇒ we can choose Γ = ZG ,l for some G s.t. F = Ψ(G ).
If F = Ψ(G ) then f and ψ(g) have the same deg d tail.
Diff (ψ(g)) ⊂ Diff (g)⇒ Zψ(g) ⊂ Γ.
Γ minimal ⇒ suff. to see that Zψ(g) apolar F
(ψ′(ψ(g)) = 0⇒ Ψ′(Ψ(G )) = 0.)
212 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
2 Parameterize the set of poly’s g ∈ K [x1, . . . xn] whose affine apolar
scheme has given length.Propostion⇐⇒ Parameterize the family of cubic
tails f ∈ K [x1, . . . , xn] of g ’s.
Cr,l =⋃
suppZl=[ld ],l(Zl )≤r 〈Zl〉
={[F ] ∈ P(Sd)|f = g≤d for some g ∈ Sloc with dim Diff (g) ≤ r}
Wr,n =⋃
l∈S1Cr,l
Cactusr (νd(P(S1))) =⋃
r1+···+rs=r J(Wr1,n, . . . ,Wrs ,n)
13 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
2 Parameterize the set of poly’s g ∈ K [x1, . . . xn] whose affine apolar
scheme has given length.Propostion⇐⇒ Parameterize the family of cubic
tails f ∈ K [x1, . . . , xn] of g ’s.
Cr,l =⋃
suppZl=[ld ],l(Zl )≤r 〈Zl〉
={[F ] ∈ P(Sd)|f = g≤d for some g ∈ Sloc with dim Diff (g) ≤ r}
Wr,n =⋃
l∈S1Cr,l
Cactusr (νd(P(S1))) =⋃
r1+···+rs=r J(Wr1,n, . . . ,Wrs ,n)
13 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
2 Parameterize the set of poly’s g ∈ K [x1, . . . xn] whose affine apolar
scheme has given length.Propostion⇐⇒ Parameterize the family of cubic
tails f ∈ K [x1, . . . , xn] of g ’s.
Cr,l =⋃
suppZl=[ld ],l(Zl )≤r 〈Zl〉
={[F ] ∈ P(Sd)|f = g≤d for some g ∈ Sloc with dim Diff (g) ≤ r}
Wr,n =⋃
l∈S1Cr,l
Cactusr (νd(P(S1))) =⋃
r1+···+rs=r J(Wr1,n, . . . ,Wrs ,n)
13 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
2 Parameterize the set of poly’s g ∈ K [x1, . . . xn] whose affine apolar
scheme has given length.Propostion⇐⇒ Parameterize the family of cubic
tails f ∈ K [x1, . . . , xn] of g ’s.
Cr,l =⋃
suppZl=[ld ],l(Zl )≤r 〈Zl〉
={[F ] ∈ P(Sd)|f = g≤d for some g ∈ Sloc with dim Diff (g) ≤ r}
Wr,n =⋃
l∈S1Cr,l
Cactusr (νd(P(S1))) =⋃
r1+···+rs=r J(Wr1,n, . . . ,Wrs ,n)
13 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Plan of the proof of the Theorem:
1 Understand the link between F and the gi ’s.
2 Parameterize the set of poly’s g ∈ K [x1, . . . xn] whose affine
apolar scheme has given length.
Propostion⇐⇒
Parameterize the family of cubic tails f ∈ K [x1, . . . , xn] of g ’s.
Find a discrete invariant for LAG schemes, parameterize the
cubic tails of all polynomials that define a scheme with given
invariant.
3 Show which invariants have the biggest family of cubic tails.
(Invariant: Hilbert function with symmetric decomposition) 14 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Plan of the proof of the Theorem:
1 Understand the link between F and the gi ’s.
2 Parameterize the set of poly’s g ∈ K [x1, . . . xn] whose affine
apolar scheme has given length.
Propostion⇐⇒
Parameterize the family of cubic tails f ∈ K [x1, . . . , xn] of g ’s.
Find a discrete invariant for LAG schemes, parameterize the
cubic tails of all polynomials that define a scheme with given
invariant.
3 Show which invariants have the biggest family of cubic tails.
(Invariant: Hilbert function with symmetric decomposition) 14 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Plan of the proof of the Theorem:
1 Understand the link between F and the gi ’s.
2 Parameterize the set of poly’s g ∈ K [x1, . . . xn] whose affine
apolar scheme has given length.
Propostion⇐⇒
Parameterize the family of cubic tails f ∈ K [x1, . . . , xn] of g ’s.
Find a discrete invariant for LAG schemes, parameterize the
cubic tails of all polynomials that define a scheme with given
invariant.
3 Show which invariants have the biggest family of cubic tails.
(Invariant: Hilbert function with symmetric decomposition) 14 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
2 Parameterize the set of poly’s g ∈ K [x1, . . . xn] whose affine
apolar scheme has given length.
Find a discrete invariant for local AG schemes, parameterize the
cubic tails of all polynomials that define a scheme with given
invariant.
f ∈ S , deg(f ) = d , f ⊥ ⊂ T
Tf := T/f ⊥∼−→τ
Diff (f ) = {ψ(f ) |ψ ∈ T}
ψ 7→ ψ(f )
15 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Iarrobino
Tf is local ⇒ one max ideal m.
m-adic filtration:
Tf = m0 ⊃ · · · ⊃ md+1 = 0
T ∗f =⊕d
i=0mi
mi+1
Loewy filtration:
Tf = (0 : md+1) ⊃ · · · ⊃
(0 : m) ⊃ 0
Interpr. in terms of partial of f
mi τ7→Partial of order at
least i of f (deg ≤ d − i)
(Order of a partial f ′ of f =
largest order of ψ ∈ T s.t.
f ′ = ψ(f ))
(0 : mi )τ7→
Diff (f )i−1 =partials of deg
at most i − 1 of f
16 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Order filtration: Diff (f ) = Diff (f )0 ⊇ Diff (f )1 ⊇ · · · ⊇ Diff (f )d
Degree filtration: Diff (f ) = Diff (f )d ⊇ Diff (f )d−1 ⊇ · · · ⊇ Diff (f )0
Different filtrations (f not homog)
but
(0 : mi )
(0 : mi−1)∼=(
mi−1
mi
)∨ Diff (f )i+1
Diff (f )i' Diff (f )i+1
Diff (f )i
and for the double filtration:
Diff (f )ai =Partials of deg at most i and order at least d − i − a
Q∨a,i 'Diff (f )ai
Diff (f )ai−1 + Diff (f )a−1i
' Diff (f )iaDiff (f )i−1
a + Diff (f )ia−1
So the Hf of the two filtrations are dual to each other.17 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
In particular
Hf (i) = dimK (Diff (f ))i − dimK (Diff (f )i−1)
has symmetric decomposition:
H =∑a≥0
∆a
each ∆a symm. around (d − a)/2, i.e. ∆a(i) = ∆a(d − a− i)
where dim(Q∨a,i ) = ∆f ,a(i).
Every partial sum∑α
a=0 ∆Qa is the Hf of a K -alg. generated in deg 1.
(Various restrictions on the possible decompositions (Mgc))
18 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Our goal was to parameterize the set of poly’s f ∈ K [x1, . . . , xn] with
a given length for their subschem Zf
Example
f = x31 + x2
2 , Length(Zf ) = 5:
1(f ) = f = x31 + x2
2
y1(f ) = x21 y2(f ) = x2
y 21 (f ) = x1 y 2
2 (f ) = 1
y 31 (f ) = 1
H = 1 2 1 1
∆0 = 1 1 1 1
∆1 = 0 1 0 0
Next step: Characterize poly’s f with given symm decomp. for Hf .
19 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Our goal was to parameterize the set of poly’s f ∈ K [x1, . . . , xn] with
a given length for their subschem Zf
Example
f = x31 + x2
2 , Length(Zf ) = 5:
1(f ) = f = x31 + x2
2
y1(f ) = x21 y2(f ) = x2
y 21 (f ) = x1 y 2
2 (f ) = 1
y 31 (f ) = 1
H = 1 2 1 1
∆0 = 1 1 1 1
∆1 = 0 1 0 0
Next step: Characterize poly’s f with given symm decomp. for Hf .
19 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
A general cubic form F ∈ S = C[y0, . . . , yn] with even local cactus
rank 2m,m ≤ n is projectively equivalent to some
f3 + x0f2 + x20 f1 + x3
0 f0
where
f3 ∈ C[x1, . . . , xm−1]3,
f2 ∈ 〈x1, ..., xn〉 · 〈x1, ..., xm−1〉,
f1 ∈ 〈x1, ..., xn〉,
f0 ∈ C.
The forms of local cactus rank 2n form a family of codimension(n−1
2
)+ 1 in the space of cubic forms C[x0, . . . , xn]3.
20 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
A general cubic form F ∈ S = C[y0, . . . , yn] with even local cactus
rank 2m,m ≤ n is projectively equivalent to some
f3 + x0f2 + x20 f1 + x3
0 f0
where
f3 ∈ C[x1, . . . , xm−1]3,
f2 ∈ 〈x1, ..., xn〉 · 〈x1, ..., xm−1〉,
f1 ∈ 〈x1, ..., xn〉,
f0 ∈ C.
The forms of local cactus rank 2n form a family of codimension(n−1
2
)+ 1 in the space of cubic forms C[x0, . . . , xn]3.
20 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
A general cubic form F ∈ S = C[y0, . . . , yn], with odd local cactus
rank 2m + 1,m ≤ n is projectively equivalent to some
f3 + xmx21 + x0f2 + x0x2
m + x20 f1 + x3
0 f0
where
f3 ∈ C[x1, . . . , xm−1]3,
f2 ∈ 〈x1, ..., xn〉 · 〈x1, ..., xm−1〉,
f1 ∈ 〈x1, ..., xn〉,
f0 ∈ C.
The forms of local cactus rank 2n + 1, n > 3 form a family of
codimension(n−2
2
)− 1 in the space of cubic forms C[x0, . . . , xn]3.
21 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
A general cubic form F ∈ S = C[y0, . . . , yn], with odd local cactus
rank 2m + 1,m ≤ n is projectively equivalent to some
f3 + xmx21 + x0f2 + x0x2
m + x20 f1 + x3
0 f0
where
f3 ∈ C[x1, . . . , xm−1]3,
f2 ∈ 〈x1, ..., xn〉 · 〈x1, ..., xm−1〉,
f1 ∈ 〈x1, ..., xn〉,
f0 ∈ C.
The forms of local cactus rank 2n + 1, n > 3 form a family of
codimension(n−2
2
)− 1 in the space of cubic forms C[x0, . . . , xn]3.
21 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
3 Show which invariants have the biggest family of cubic tails.
Cr ,l =⋃
supp(Zl )=ν3([l ]),l(Zl )≤r 〈Zl〉 ⇔ Parameterize the family of cubic
tails f ∈ K [x1, . . . , xn] of g ’s:
Vr (3, n) = {f | f = g≤3 for some g ∈ K [x1, . . . , xn], dim Diff (g) ≤ r}
V (3,∆, n) = {f≤3 | f ∈ K [x1, . . . , xn],∆f = ∆}
Vr (3, n) =⋃
l(∆)≤r
V (3,∆, n)
For which Hf and which ∆, v(3,∆, n) := dim(V (3,∆, n)) attains its
max given r? =⇒ max(v(3,∆, n)) =upper bound for
vr (3, n) := dim(Vr (3,∆))22 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
3 Show which invariants have the biggest family of cubic tails.
Cr ,l =⋃
supp(Zl )=ν3([l ]),l(Zl )≤r 〈Zl〉 ⇔ Parameterize the family of cubic
tails f ∈ K [x1, . . . , xn] of g ’s:
Vr (3, n) = {f | f = g≤3 for some g ∈ K [x1, . . . , xn], dim Diff (g) ≤ r}
V (3,∆, n) = {f≤3 | f ∈ K [x1, . . . , xn],∆f = ∆}
Vr (3, n) =⋃
l(∆)≤r
V (3,∆, n)
For which Hf and which ∆, v(3,∆, n) := dim(V (3,∆, n)) attains its
max given r? =⇒ max(v(3,∆, n)) =upper bound for
vr (3, n) := dim(Vr (3,∆))22 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
3 Show which invariants have the biggest family of cubic tails.
Cr ,l =⋃
supp(Zl )=ν3([l ]),l(Zl )≤r 〈Zl〉 ⇔ Parameterize the family of cubic
tails f ∈ K [x1, . . . , xn] of g ’s:
Vr (3, n) = {f | f = g≤3 for some g ∈ K [x1, . . . , xn], dim Diff (g) ≤ r}
V (3,∆, n) = {f≤3 | f ∈ K [x1, . . . , xn],∆f = ∆}
Vr (3, n) =⋃
l(∆)≤r
V (3,∆, n)
For which Hf and which ∆, v(3,∆, n) := dim(V (3,∆, n)) attains its
max given r? =⇒ max(v(3,∆, n)) =upper bound for
vr (3, n) := dim(Vr (3,∆))22 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Proposition
r ≥ 7, v(3,∆, n) attains its max for ∆ = (1,m − 1,m − 1, 1), r = 2m,
∆ = (1, 1, 1, 1, 1), (0,m − 2,m − 2, 0), r = 2m + 1
and
v(3,∆, n) =
Me :=(m+2
3
)+ 2m(n −m) + 3m − n − 1, r = 2m,
Mo :=(m+2
3
)+ 2m(n −m) + 3m − 2, r = 2m + 1.
23 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Cactusr (X3,n) =⋃
r1+···+rsJ(Wr1,n, . . . ,Wrs ,n)
Wr ,n =⋃
l∈S1Cr ,l Local cactus variety
Cr ,l = Vr (3, n) =⋃
l(∆)≤r V (3,∆, n)
max(v(3,∆, n))⇒ upper bound for vr (3, n), i.e. for dim(Cr ,l)
The largest component of W2m,n is the union as l varies, of projective
varieties whose affine cones are isomorphic to V2m(3, n), so a
parameterization of W2m,n has dimension v2m(3, n)−1 + n.
Similarly, the largest component of W2m+1,n is the union as l varies,
of varieties isomorphic to V2m+1(3, n), so a parameterization of
W2m+1,n has dimension v2m+1(3, n)+n.
BUT parameterization may not be generically finite, the formulas are
upper bounds for the dimension of these varieties.24 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Cactusr (X3,n) =⋃
r1+···+rsJ(Wr1,n, . . . ,Wrs ,n)
Wr ,n =⋃
l∈S1Cr ,l Local cactus variety
Cr ,l = Vr (3, n) =⋃
l(∆)≤r V (3,∆, n)
max(v(3,∆, n))⇒ upper bound for vr (3, n), i.e. for dim(Cr ,l)
The largest component of W2m,n is the union as l varies, of projective
varieties whose affine cones are isomorphic to V2m(3, n), so a
parameterization of W2m,n has dimension v2m(3, n)−1 + n.
Similarly, the largest component of W2m+1,n is the union as l varies,
of varieties isomorphic to V2m+1(3, n), so a parameterization of
W2m+1,n has dimension v2m+1(3, n)+n.
BUT parameterization may not be generically finite, the formulas are
upper bounds for the dimension of these varieties.24 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Cactusr (X3,n) =⋃
r1+···+rsJ(Wr1,n, . . . ,Wrs ,n)
Wr ,n =⋃
l∈S1Cr ,l Local cactus variety
Cr ,l = Vr (3, n) =⋃
l(∆)≤r V (3,∆, n)
max(v(3,∆, n))⇒ upper bound for vr (3, n), i.e. for dim(Cr ,l)
The largest component of W2m,n is the union as l varies, of projective
varieties whose affine cones are isomorphic to V2m(3, n), so a
parameterization of W2m,n has dimension v2m(3, n)−1 + n.
Similarly, the largest component of W2m+1,n is the union as l varies,
of varieties isomorphic to V2m+1(3, n), so a parameterization of
W2m+1,n has dimension v2m+1(3, n)+n.
BUT parameterization may not be generically finite, the formulas are
upper bounds for the dimension of these varieties.24 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Cactusr (X3,n) =⋃
r1+···+rsJ(Wr1,n, . . . ,Wrs ,n)
Wr ,n =⋃
l∈S1Cr ,l Local cactus variety
Cr ,l = Vr (3, n) =⋃
l(∆)≤r V (3,∆, n)
max(v(3,∆, n))⇒ upper bound for vr (3, n), i.e. for dim(Cr ,l)
The largest component of W2m,n is the union as l varies, of projective
varieties whose affine cones are isomorphic to V2m(3, n), so a
parameterization of W2m,n has dimension v2m(3, n)−1 + n.
Similarly, the largest component of W2m+1,n is the union as l varies,
of varieties isomorphic to V2m+1(3, n), so a parameterization of
W2m+1,n has dimension v2m+1(3, n)+n.
BUT parameterization may not be generically finite, the formulas are
upper bounds for the dimension of these varieties.24 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Cactusr (X3,n) =⋃
r1+···+rsJ(Wr1,n, . . . ,Wrs ,n)
Wr ,n =⋃
l∈S1Cr ,l Local cactus variety
Cr ,l = Vr (3, n) =⋃
l(∆)≤r V (3,∆, n)
max(v(3,∆, n))⇒ upper bound for vr (3, n), i.e. for dim(Cr ,l)
The largest component of W2m,n is the union as l varies, of projective
varieties whose affine cones are isomorphic to V2m(3, n), so a
parameterization of W2m,n has dimension v2m(3, n)−1 + n.
Similarly, the largest component of W2m+1,n is the union as l varies,
of varieties isomorphic to V2m+1(3, n), so a parameterization of
W2m+1,n has dimension v2m+1(3, n)+n.
BUT parameterization may not be generically finite, the formulas are
upper bounds for the dimension of these varieties.24 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Cactusr (X3,n) =⋃
r1+···+rsJ(Wr1,n, . . . ,Wrs ,n)
Wr ,n =⋃
l∈S1Cr ,l Local cactus variety
Cr ,l = Vr (3, n) =⋃
l(∆)≤r V (3,∆, n)
max(v(3,∆, n))⇒ upper bound for vr (3, n), i.e. for dim(Cr ,l)
The largest component of W2m,n is the union as l varies, of projective
varieties whose affine cones are isomorphic to V2m(3, n), so a
parameterization of W2m,n has dimension v2m(3, n)−1 + n.
Similarly, the largest component of W2m+1,n is the union as l varies,
of varieties isomorphic to V2m+1(3, n), so a parameterization of
W2m+1,n has dimension v2m+1(3, n)+n.
BUT parameterization may not be generically finite, the formulas are
upper bounds for the dimension of these varieties.24 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Cactusr (X3,n) =⋃
r1+···+rsJ(Wr1,n, . . . ,Wrs ,n)
Wr ,n =⋃
l∈S1Cr ,l Local cactus variety
Cr ,l = Vr (3, n) =⋃
l(∆)≤r V (3,∆, n)
max(v(3,∆, n))⇒ upper bound for vr (3, n), i.e. for dim(Cr ,l)
The largest component of W2m,n is the union as l varies, of projective
varieties whose affine cones are isomorphic to V2m(3, n), so a
parameterization of W2m,n has dimension v2m(3, n)−1 + n.
Similarly, the largest component of W2m+1,n is the union as l varies,
of varieties isomorphic to V2m+1(3, n), so a parameterization of
W2m+1,n has dimension v2m+1(3, n)+n.
BUT parameterization may not be generically finite, the formulas are
upper bounds for the dimension of these varieties.24 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
Cactusr (X3,n) =⋃
r1+···+rsJ(Wr1,n, . . . ,Wrs ,n)
Wr ,n =⋃
l∈S1Cr ,l Local cactus variety
Cr ,l = Vr (3, n) =⋃
l(∆)≤r V (3,∆, n)
max(v(3,∆, n))⇒ upper bound for vr (3, n), i.e. for dim(Cr ,l)
The largest component of W2m,n is the union as l varies, of projective
varieties whose affine cones are isomorphic to V2m(3, n), so a
parameterization of W2m,n has dimension v2m(3, n)−1 + n.
Similarly, the largest component of W2m+1,n is the union as l varies,
of varieties isomorphic to V2m+1(3, n), so a parameterization of
W2m+1,n has dimension v2m+1(3, n)+n.
BUT parameterization may not be generically finite, the formulas are
upper bounds for the dimension of these varieties.24 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
We have equality if we show that the parameterization is generically
one to one.
When b is even, for a general [F ] ∈Wb,n there is a unique l such
that ZF ,l has length b.
When b is odd, and there is a unique l such that f = πl(F ) is the tail
of a quartic polynomial gl whose apolar scheme Zgl has length b.
So dim(Cactusr (X3,n)) = dim Wr ,n.
25 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
We have equality if we show that the parameterization is generically
one to one.
When b is even, for a general [F ] ∈Wb,n there is a unique l such
that ZF ,l has length b.
When b is odd, and there is a unique l such that f = πl(F ) is the tail
of a quartic polynomial gl whose apolar scheme Zgl has length b.
So dim(Cactusr (X3,n)) = dim Wr ,n.
25 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
We have equality if we show that the parameterization is generically
one to one.
When b is even, for a general [F ] ∈Wb,n there is a unique l such
that ZF ,l has length b.
When b is odd, and there is a unique l such that f = πl(F ) is the tail
of a quartic polynomial gl whose apolar scheme Zgl has length b.
So dim(Cactusr (X3,n)) = dim Wr ,n.
25 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
Proof of the Thm
We have equality if we show that the parameterization is generically
one to one.
When b is even, for a general [F ] ∈Wb,n there is a unique l such
that ZF ,l has length b.
When b is odd, and there is a unique l such that f = πl(F ) is the tail
of a quartic polynomial gl whose apolar scheme Zgl has length b.
So dim(Cactusr (X3,n)) = dim Wr ,n.
25 / 26
Cactus
Varieties of
Cubic Forms:
Apolar Local
Artinian
Gorenstein
Rings
[–, J.Jelisiejew,P. MaciasMarques,
K.Ranestad]
Alessandra
Bernardi
THANKS!
26 / 26