Notre Dame Algebraic Geometry Seminar

Spring 2008

 

Date

Time/Place

Speaker

Title

Jan 25

258 Hurley

4pm

C. Polini

(Notre Dame)

Integrality, Rees ring, and adjoint ideals (I)

Feb 1

258 Hurley

4pm

C. Polini

(Notre Dame)

Integrality, Rees ring, and adjoint ideals (II)

Feb 8

258 Hurley

4pm

J. Migliore

(Note Dame)

Gorenstein Hilbert functions

Feb 29

258 Hurley

4pm

C. Polini

(Notre Dame)

Integrality, Rees ring, and adjoint ideals (III)

Mar 14

258 Hurley

4pm

F. Zanello

 

(Michigan Tech)

Interval Conjectures for level Hilbert functions

Apr 7

258 Hurley

4pm

R. Hartshorne

 

(Berkeley)

Ulrich bundles on cubic surfaces

Apr 18

258 Hurley

4pm

C. Polini

(Notre Dame)

Integrality, Rees ring, and adjoint ideals (IV)

Apr 25

258 Hurley

4pm

C.Polini

(Notre Dame)

Integrality, Rees ring, and adjoint ideals (V)

May 21

 

T. Szemberg

(Krakow)

Geometric obstructions for
Seshadri constants

May 21

 

H. Maeda

(Waseda)

TBA


Abstract of Talks:

Feb 8: J. Migliore (Notre Dame).  Gorenstein Hilbert functions.

What are the possible Hilbert functions of graded Artinian Gorenstein algebras?  Many authors have studied this question.  When the algebras have the Weak Lefschetz Property (WLP), the Hilbert functions are the so-called Stanley-Iarrobino (SI) sequences.  This implies, in particular, that they are unimodal.  In codimension 3, the possible Hilbert functions are again precisely the SI-sequences, even though it is still an open question whether all such algebras have WLP.  In codimension five or more, it is known that non-unimodal examples exist.  We will present some asymptotic results describing "how non-unimodal" they can be, and in the process answer an old conjecture of Richard Stanley.  This leaves codimension 4 hanging in the balance.  It is not known whether non-unimodal, or even non-SI, examples exist, although it is known that not all such algebras have WLP.  We will present recent work showing that at least for low initial degree, all occurring Hilbert functions are SI-sequences (hence unimodal).  All new work described is joint with Uwe Nagel and Fabrizio Zanello.


Mar 14: F. Zanello (Michigan Tech). Interval Conjectures for level Hilbert functions.

Abstract: The theory of Gorenstein and level (graded) algebras is an important topic of commutative algebra, because of both its intrinsic interest and the applications it has to several other fields - such as algebraic combinatorics, algebraic geometry, invariant theory, and even complexity theory.
 One fundamental invariant of graded algebras is the Hilbert function, which counts the dimension of such algebras in each graded piece.
 The goal of this talk is to present and discuss two conjectures I have recently formulated: The "Interval Conjecture" (IC) and the "Gorenstein Interval Conjecture" (GIC).
 These conjectures are inspired by the research performed in this area over the last few years. In particular, a series of recent results seems to indicate that it is nearly impossible to characterize explicitly the sets of all Gorenstein or of level Hilbert functions. Therefore, the purpose of the IC and the GIC is to at least provide the existence of a very strong - and natural - form of "regularity" in the structure of such important and complicated sets.
 Even if I have already proved a few particular cases, we still seem very far from showing my conjectures in full generality at this point.
 In this talk I will also discuss the background and the main results obtained so far in this area, as well as the techniques I have employed to begin studying the two conjectures.

REFERENCES:
A Iarrobino: Hilbert functions of Gorenstein algebras associated to a pencil of forms. Projective varieties with unexpected properties, 273--286, Walter de Gruyter GmbH & Co. KG, Berlin, 2005.
F. Z.: Partial derivatives of a generic subspace of a vector space of forms: quotients of level algebras of arbitrary type, Trans. of the A.M.S. 359 (2007), No. 2, 2675-2686.
F. Z.: Interval Conjectures for level Hilbert functions, J. of Algebra, to appear.

 

April 7: R. Hartshorne (Berkeley). Ulrich bundles on cubic surfaces.

Abstract: Let X be a nonsingular cubic surface in P^3. We prove the existence, for every rank r at least 2, of stable Ulrich bundles of rank r on X. An Ulrich bundle is a vector bundle with no intermediate cohomology, whose associated graded module

has the largest possible number, 3r, of generators. These bundles form an open subset of dimension r^2 + 1 of the moduli space of stable vector bundles on X.

 

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