The Weak Lefschetz for a Graded Module Zachary Flores 1/24 Basic - - PowerPoint PPT Presentation

Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

The Weak Lefschetz for a Graded Module

Zachary Flores

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Solomon Lefschetz

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Solomon Lefschetz

1 Lefschetz lost both his hands in an engineering accident

and subsequently became a mathematician.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Solomon Lefschetz

1 Lefschetz lost both his hands in an engineering accident

and subsequently became a mathematician.

2 Lefschtez was an instructor at UNL from 1911-1913.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Solomon Lefschetz

1 Lefschetz lost both his hands in an engineering accident

and subsequently became a mathematician.

2 Lefschtez was an instructor at UNL from 1911-1913. 3 Lefschetz was a professor at KU from 1913-1924.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Solomon Lefschetz

1 Lefschetz once received the following letter of

recommendation for John Nash.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Solomon Lefschetz

1 Lefschetz once received the following letter of

recommendation for John Nash.

2 My roommate once held the door open for John Nash.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Solomon Lefschetz

1 Lefschetz once received the following letter of

recommendation for John Nash.

2 My roommate once held the door open for John Nash. 3 Conclusion: I have met Solomon Lefschetz.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Outline

1 Basic Concepts 2 Semistable Bundles 3 Symmetric Hilbert Functions 4 Main Theorem

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

The Weak Lefschetz Property

Let k be an algebraically closed field and S the polynomial ring k[x1, . . . , xr].

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

The Weak Lefschetz Property

Let k be an algebraically closed field and S the polynomial ring k[x1, . . . , xr]. Definition Given a graded S-module N of finite length, we say that N has the Weak Lefschetz Property if for any general linear form ℓ ∈ S1, the map ×ℓ : Nt → Nt+1 has maximal rank for all t.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

The Weak Lefschetz Property

Let k be an algebraically closed field and S the polynomial ring k[x1, . . . , xr]. Definition Given a graded S-module N of finite length, we say that N has the Weak Lefschetz Property if for any general linear form ℓ ∈ S1, the map ×ℓ : Nt → Nt+1 has maximal rank for all t. Question: Which graded S-modules of finite length have the Weak Lefschetz Property?

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

The Weak Lefschetz Property

When N = S/I with I a homogeneous ideal of codimension r, the Weak Lefschetz Property has been studied extensively.

1 When r ≤ 2, S/I always has the Weak Lefschetz.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

The Weak Lefschetz Property

When N = S/I with I a homogeneous ideal of codimension r, the Weak Lefschetz Property has been studied extensively.

1 When r ≤ 2, S/I always has the Weak Lefschetz. 2 In [1] it is shown that if k has characteristic 0, r = 3 and I

is a complete intersection generated in certain degrees, then S/I has the Weak Lefschetz. This was used to prove all complete intersections in three variables have the Weak Lefschetz.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

The Weak Lefschetz Property

When N = S/I with I a homogeneous ideal of codimension r, the Weak Lefschetz Property has been studied extensively.

1 When r ≤ 2, S/I always has the Weak Lefschetz. 2 In [1] it is shown that if k has characteristic 0, r = 3 and I

is a complete intersection generated in certain degrees, then S/I has the Weak Lefschetz. This was used to prove all complete intersections in three variables have the Weak Lefschetz. What about characteristic p > 0?

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

An Example

Suppose r = 3, k has characteristic 2 and I = (x2

1, x2 2, x2 3). If

ℓ = ax1 + bx2 + cx3 is a linear form, then a matrix for the map ×ℓ : (S/I)1 → (S/I)2 is given by A =   b a c a c b  

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

An Example

Suppose r = 3, k has characteristic 2 and I = (x2

1, x2 2, x2 3). If

ℓ = ax1 + bx2 + cx3 is a linear form, then a matrix for the map ×ℓ : (S/I)1 → (S/I)2 is given by A =   b a c a c b   Then det(A) = −2abc = 0, so ×ℓ is not injective. Thus S/I does not have the Weak Lefschetz.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

An Example

Suppose r = 3, k has characteristic 2 and I = (x2

1, x2 2, x2 3). If

ℓ = ax1 + bx2 + cx3 is a linear form, then a matrix for the map ×ℓ : (S/I)1 → (S/I)2 is given by A =   b a c a c b   Then det(A) = −2abc = 0, so ×ℓ is not injective. Thus S/I does not have the Weak Lefschetz. In fact, if r ≥ 3 and k has characteristic p > 0, then S/(xp

1, . . . , xp r) does not have the Weak Lefschetz.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Setup

As the above results suggest, the characteristic of k plays a subtle role in determining when N has the Weak Lefschetz.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Setup

As the above results suggest, the characteristic of k plays a subtle role in determining when N has the Weak Lefschetz.

1 k has characteristic zero (and is still algebraically closed!).

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Setup

As the above results suggest, the characteristic of k plays a subtle role in determining when N has the Weak Lefschetz.

1 k has characteristic zero (and is still algebraically closed!). 2 We set R = k[x, y, z].

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Setup

As the above results suggest, the characteristic of k plays a subtle role in determining when N has the Weak Lefschetz.

1 k has characteristic zero (and is still algebraically closed!). 2 We set R = k[x, y, z]. 3 n ≥ 1, ϕ : n+2 j=1 R(−bj) → n i=1 R(−ai) is an R-linear

map with bj ≤ bj+1 and ai ≤ ai+1.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Setup

As the above results suggest, the characteristic of k plays a subtle role in determining when N has the Weak Lefschetz.

1 k has characteristic zero (and is still algebraically closed!). 2 We set R = k[x, y, z]. 3 n ≥ 1, ϕ : n+2 j=1 R(−bj) → n i=1 R(−ai) is an R-linear

map with bj ≤ bj+1 and ai ≤ ai+1.

4 We set M = coker(ϕ).

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Setup

As the above results suggest, the characteristic of k plays a subtle role in determining when N has the Weak Lefschetz.

1 k has characteristic zero (and is still algebraically closed!). 2 We set R = k[x, y, z]. 3 n ≥ 1, ϕ : n+2 j=1 R(−bj) → n i=1 R(−ai) is an R-linear

map with bj ≤ bj+1 and ai ≤ ai+1.

4 We set M = coker(ϕ).

When M has finite length, we are interested in what numerical constraints we can place on the bj and ai so that M has the Weak Lefschetz.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Some Algebraic Geometry

If E is a vector bundle on Pr, its slope µ(E) is the rational number c1(E)/rank(E), where c1(E) is the first Chern class of E.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Some Algebraic Geometry

If E is a vector bundle on Pr, its slope µ(E) is the rational number c1(E)/rank(E), where c1(E) is the first Chern class of E. Definition We say that a vector bundle E on Pr is semistable if µ(E′) ≤ µ(E) for every proper nonzero subbundle E′ of E.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Some Algebraic Geometry

If E is a vector bundle on Pr, its slope µ(E) is the rational number c1(E)/rank(E), where c1(E) is the first Chern class of E. Definition We say that a vector bundle E on Pr is semistable if µ(E′) ≤ µ(E) for every proper nonzero subbundle E′ of E. For E with rank 2 and E normalized (c1(E) ∈ {−1, 0}), E is semistable if and only if it has no sections.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Some Algebraic Geometry

By a theorem of Grothendieck, every vector bundle on P1 splits as a sum of line bundles. Hence, if λ is general line in P2 and E is a vector bundle on P2 then E|λ = Oλ(e1) ⊕ · · · ⊕ Oλ(es)

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Some Algebraic Geometry

By a theorem of Grothendieck, every vector bundle on P1 splits as a sum of line bundles. Hence, if λ is general line in P2 and E is a vector bundle on P2 then E|λ = Oλ(e1) ⊕ · · · ⊕ Oλ(es) Where the ek are independent of λ and s = rank(E). The s-tuple (e1, . . . , es) is called the splitting type of E.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

The Grauert-M¨ ulich Theorem

Theorem If E is a semistable normalized vector bundle of rank two on P2 and λ is a general line in P2, then (e1, e2) = (0, 0) c1(E) = 0 (0, −1) c1(E) = −1

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

The Grauert-M¨ ulich Theorem

Theorem If E is a semistable normalized vector bundle of rank two on P2 and λ is a general line in P2, then (e1, e2) = (0, 0) c1(E) = 0 (0, −1) c1(E) = −1 We will see how this theorem plays a crucial role in determining when M has the Weak Lefschetz.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

The Minimal Free Resolution of M

M has finite length if and only if its ideal of maximal minors has codimension 3. In this case, the Buchsbaum-Rim complex gives the minimal free resolution for M:

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

The Minimal Free Resolution of M

M has finite length if and only if its ideal of maximal minors has codimension 3. In this case, the Buchsbaum-Rim complex gives the minimal free resolution for M: 0 → F3 → F2 →

n+2

• j=1

R(−bj)

ϕ

→

n

• i=1

R(−ai) Where F3 = n

i=1 R(ai − d) and F2 = m j=1 R(bj − d) with

d = bj − ai.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Bundles from Syzygies

Set E = ker(ϕ), so that upon sheafification, we obtain the following exact sequence of sheaves over P2: 0 → F3 → F2 → E → 0 Thus E is a vector bundle of rank two and we want to know when E is semistable.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Bundles from Syzygies

Set E = ker(ϕ), so that upon sheafification, we obtain the following exact sequence of sheaves over P2: 0 → F3 → F2 → E → 0 Thus E is a vector bundle of rank two and we want to know when E is semistable. Lemma Suppose n > 1. Then E is semistable when (a) d is even,

i<n bi − ai > −4, an + bm + 1 < bn + bn+1

(b) d is odd,

i<n bi − ai > −4, an + bm + 2 < bn + bn+1

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Bundles from Syzygies

A twist combined with the Grauert-M¨ ulich Theorem yields Corollary If n > 1 and either condition of the above lemma is satisfied, then the splitting type (e1, e2) of E is (e1, e2) =

• (−e, −e)

d = 2e (−e, −e − 1) d = 2e + 1

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Bundles from Syzygies

A twist combined with the Grauert-M¨ ulich Theorem yields Corollary If n > 1 and either condition of the above lemma is satisfied, then the splitting type (e1, e2) of E is (e1, e2) =

• (−e, −e)

d = 2e (−e, −e − 1) d = 2e + 1 With ([1], Lemma 2.1, Corollary 2.2), we have numerical conditions for all n on the ai and bj that tell us when E is semistable and, if that is the case, what its splitting type is.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Symmetrically Gorenstein Modules

In [1], one of the key ingredients in proving that complete intersections of codimension 3 have the Weak Lefschetz Property was the symmetry of the Hilbert function. It is a well-known fact that graded Gorenstein k-algebras have a symmetric Hilbert function.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Symmetrically Gorenstein Modules

In [1], one of the key ingredients in proving that complete intersections of codimension 3 have the Weak Lefschetz Property was the symmetry of the Hilbert function. It is a well-known fact that graded Gorenstein k-algebras have a symmetric Hilbert function. Definition We say a graded S-module N of finite length is Symmetrically Gorenstein if there is a v ∈ Z and a graded isomorphism τ : N → Homk(N, k)(−v) such that τ = Homk(τ, k)(−v).

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Kunte’s Theorem

When is M Symmetrically Gorenstein?

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Kunte’s Theorem

When is M Symmetrically Gorenstein? We have the following special case of ([2], Theorem 1.3). Theorem Let N be graded R-module of finite length and of maximal socle degree c. Set (•)∨ = HomR(•, R(−c − 3)). Then N is Symmetrically Gorenstein if and only if its minimal graded free resolution is of the form 0 → G∨

0 → G∨ 1 Ψ

→ G1 → G0 where Ψ is antisymmetric.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Kunte’s Theorem

When is M Symmetrically Gorenstein? We have the following special case of ([2], Theorem 1.3). Theorem Let N be graded R-module of finite length and of maximal socle degree c. Set (•)∨ = HomR(•, R(−c − 3)). Then N is Symmetrically Gorenstein if and only if its minimal graded free resolution is of the form 0 → G∨

0 → G∨ 1 Ψ

→ G1 → G0 where Ψ is antisymmetric. It is not hard to see that under mild conditions, Symmetrically Gorenstein modules have symmetric Hilbert functions.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Using Kunte’s Theorem

Lemma M has a symmetric Hilbert function if a1 = 0 and its maximal socle degree is d − 3.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Using Kunte’s Theorem

Lemma M has a symmetric Hilbert function if a1 = 0 and its maximal socle degree is d − 3. When a1 = 0, M will have a maximal socle degree d − 3 under reasonable assumptions.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Main Theorem

Theorem If E is semistable and M has symmetric Hilbert function, then M has the Weak Lefschetz Property.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Main Theorem

Theorem If E is semistable and M has symmetric Hilbert function, then M has the Weak Lefschetz Property. In particular, utilizing our preceding work, we have numerical conditions on the ai and bj that guarantee that M has the Weak Lefschetz.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Main Theorem

Theorem If E is semistable and M has symmetric Hilbert function, then M has the Weak Lefschetz Property. In particular, utilizing our preceding work, we have numerical conditions on the ai and bj that guarantee that M has the Weak Lefschetz. Corollary Complete intersections in R have the Weak Lefschetz Property.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Sketch of proof

Recall ϕ : n+2

j=1 R(−bj) → n i=1 R(−ai) is an R-linear map

with cokernel M.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Sketch of proof

Recall ϕ : n+2

j=1 R(−bj) → n i=1 R(−ai) is an R-linear map

with cokernel M. Proof. Let ℓ be a general linear form in R and λ be the general line defined by ℓ in P2. After sheafifying, some diagram chasing and an application of the Snake Lemma, we obtain following exact sequence of sheaves:

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Sketch of proof

Recall ϕ : n+2

j=1 R(−bj) → n i=1 R(−ai) is an R-linear map

with cokernel M. Proof. Let ℓ be a general linear form in R and λ be the general line defined by ℓ in P2. After sheafifying, some diagram chasing and an application of the Snake Lemma, we obtain following exact sequence of sheaves: 0 → E|λ →

n+2

• j=1

Oλ(−bj) →

n

• i=1

Oλ(−ai) → 0 (⋆)

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Sketch of proof

Recall ϕ : n+2

j=1 R(−bj) → n i=1 R(−ai) is an R-linear map

with cokernel M. Proof. Let ℓ be a general linear form in R and λ be the general line defined by ℓ in P2. After sheafifying, some diagram chasing and an application of the Snake Lemma, we obtain following exact sequence of sheaves: 0 → E|λ →

n+2

• j=1

Oλ(−bj) →

n

• i=1

Oλ(−ai) → 0 (⋆) When d = 2e, the semistability of E gives, by our previous work, E|λ = Oλ(−e)2.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Proof. Set R = R/ℓR. If N = im(ϕ) ⊆ n

i=1 R(−ai), taking global

sections in (⋆) yields the exact sequence of R-modules:

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Proof. Set R = R/ℓR. If N = im(ϕ) ⊆ n

i=1 R(−ai), taking global

sections in (⋆) yields the exact sequence of R-modules: 0 → R(−e)2 →

n+2

• j=1

R(−bj)

ϕ

→ N → 0 (⋆⋆)

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Proof. Set R = R/ℓR. If N = im(ϕ) ⊆ n

i=1 R(−ai), taking global

sections in (⋆) yields the exact sequence of R-modules: 0 → R(−e)2 →

n+2

• j=1

R(−bj)

ϕ

→ N → 0 (⋆⋆) Since the Hilbert function is symmetric, it suffices to show that for u ≤ ⌊ d−3

2 ⌋ = e − 2, Mu ×ℓ

→ Mu+1 is injective.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Proof. To this end, if ℓ kills a nontrivial element of Mu, there is an equation ℓF = ϕA with F ∈ n

i=1 R(−ai) and

A ∈ n+2

j=1 R(−bj).

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Proof. To this end, if ℓ kills a nontrivial element of Mu, there is an equation ℓF = ϕA with F ∈ n

i=1 R(−ai) and

A ∈ n+2

j=1 R(−bj).

Reducing modulo ℓ, we obtain a nontrivial element A of ker(ϕ). Using (⋆⋆), it is not hard to see that this would yield a syzygy

• f ϕ that has degree u + 1 − e, which is impossible. When d is
• dd, the proof is similar.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Example

Example Let n = 2, q ≥ 3 and f1, f2, f3 be a regular sequence in Rq and ϕ : R(−q)4 → R2 defined by f1 f2 f3 f1 f2 f3

• Then M = coker(ϕ) has the Weak Lefschetz and its minimal

number of generators as an R-module is 2.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

References

1 Harima, T., Migliore, J.C., Nagel, U., Watanabe, J., The

Weak and Strong Lefschetz Properties for Artinian K-algebras. Journal of Algebra 262(1), pp. 99-126, 2003.

2 Kunte, M., Gorenstein Modules of Finite Length.

Mathematische Nachrichten 284(7), pp. 899-919, 2011.

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Basic Concepts Semistable Bundles Symmetric Hilbert Functions Main Theorem

Acknowledgements

• A big thanks to the organizers for inviting me to speak.
• I would like to thank Chris Peterson for suggesting this

problem, his helpful comments and encouragement on this project.

• I would also like to thank Gioia Failla for her helpful

comments in preparing the paper this talk is based on.

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