Algebraic Geometry/Commutative Algebra Seminar, 2013-2014

To volunteer to give a talk, or for anything else regarding the seminar, contact Juan Migliore.

Spring Schedule

The seminar will meet on Wednesdays, 3:00-4:00 in 258 Hurley unless otherwise noted.

Fall Schedule

Abstracts

Friday, September 13, 2013, 3:30-4:30

Speaker

Sam Evens (Notre Dame)

Title

Eigenvalue coincidences and orbits on the flag variety

Abstract

Click here.

Friday, November 1, 2013, 3:30-4:30

Speaker

David Cook II (Notre Dame)

Title

An introduction to Boij-Soederberg theory

Abstract

Boij-Soederberg theory describes the Betti tables of standard graded modules over the polynomial ring, up to scaling by a rational number. In particular, it provides an algorithm to decompose the Betti table of a standard graded module into a positive linear combination of pure Betti tables, i.e., Betti tables that have only one entry per column.

Friday, November 8, 2013, 3:30-4:30

Speaker

Benjamin Bakker (NYU)

Title

On the Frey-Mazur conjecture over low genus curves

Abstract

A crucial step in the proof of Fermat's last theorem was Frey's insight that a nontrivial solution would yield an elliptic curve with modular p-torsion but which was itself not modular. The connection between an elliptic curve and its p-torsion is very deep: a conjecture of Frey and Mazur stating that p-torsion actually determines the elliptic curve up to isogeny (at least when p>13) implies an asymptotic generalization of Fermat's last theorem. We study a geometric analog of this conjecture, and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic multiplication---to their p-torsion Galois representations is at most two-to-one, and one-to-one in special cases. Our proof fundamentally uses the interaction between the hyperbolic and algebraic properties of Shimura varieties. This is joint work with Jacob Tsimerman.

Friday, November 15, 2013, 3:30-4:30

Speaker

Robin Hartshorne (Berkeley)

Title

Set-theoretic complete intersections and divisor class groups

Abstract

I will begin by reviewing some of the history of the problem of set-theoretic complete intersections in projective space. Then I will report on some recent work, joint with Claudia Polini, inspired by this problem, which provides a small contribution to the problem itself.

Friday, December 6, 2013, 3:30-4:30

Speaker

Andrei Jorza (Notre Dame)

Title

Lagrangian planes in holomorphic symplectic varieties

Abstract

For a K3 surface X the cone of effective curve classes on X has extremal rays given by the rational curves C such that (C,C)=-2. It is natural to ask whether a similar description exists for higher dimensional holomorphic symplectic varieties X. The intersection form is no longer a quadratic form on curve classes, but the Beauville-Bogomolov form on the cohomology of X induces a nondegenerate form (,) on H_2(X) which coincides with the intersection pairing when X is a K3 surface. A desirable characterization of extremal rays in the effective cone of X would be that they are precisely the rational curves C such that (C,C)=-c for some positive rational number c. The class of any line L contained in a Lagrangian plane inside X will be extremal and, when X is deformation equivalent to the Hilbert scheme of n points on a K3 surface, Hassett and Tschinkel conjectured that (L, L) = -(n+3)/2. In joint work with Benjamin Bakker we proved this conjecture when n=4 by viewing (L,L) as a coordinate of a rational point on a specific curve.

Wednesday, January 15, 2014, 3:00-4:00

Speaker

Botong Wang (Notre Dame)

Title

A conjecture of Beauville and Catanese for compact Kahler manifolds

Abstract

Click here.

Wednesday, January 22, 2014, 3:00-4:00

Speaker

Moshe Kamensky (Hebrew University)

Title

Tannakian formalism for groups over fields with operators

Abstract

The classical Tannakian formalism describes the structure of categories of the form Rep_G, the category of representations of an affine group scheme G over a field. When the base field is endowed with additional structure, such a derivation or an automorphism, it is possible to consider more general groups, defined by differential (or difference) equations, rather than polynomial ones. I will explain a general framework of ``fields with operators'' in which one could consider such groups, and the generalisation of the Tannakian formalism to this setting.

Wednesday, February 19, 2014, 3:00-4:00

Speaker

Julie Déserti (University of Paris 7)

Title

Birational maps of P^3_C of small bidegree

Abstract

Click here.

Wednesday, February 26 and March 5, 2014, 3:00-4:00

Speaker

Jeff Madsen (Notre Dame)

Title

Rees algebras of parameterized plane curves

Abstract

If C is a rational parameterized plane curve of degree d, the bihomogeneous coordinate ring of its graph is given by the Rees algebra of an almost complete intersection ideal in k[x,y]. The Rees algebra can be viewed as the quotient of the symmetric algebra by its torsion ideal A. Finding a minimal generating set of A is largely an open problem, though it has been solved, for instance, for d<=6 by the work of Busé and of Kustin, Polini, and Ulrich. I will present results that can be used to find all possible bidegrees of the minimal generators of A when d=7, and show how these degrees correspond to the singularities of C.

Wednesday, March 26, 2014, 3:00-4:00

Speaker

Andrei Jorza (Notre Dame)

Title

Counting sheaves on surfaces

Abstract

There is a rich theory of counting curves on manifolds which bridges algebraic geometry and physics. In the case of threefolds, Gromov-Witten theory formalized certain curve counts first obtained (correctly) by physicists, by expressing such curve counts as integrals with respect to a virtual class on a certain moduli space. Donaldson-Thomas theory and Pandharipande-Thomas theory produced frameworks for counting sheaves on threefolds, both conjecturally equivalent to Gromov-Witten theory.

The analogous picture for surfaces is incomplete. The Gromov-Witten theory has been calculated by Maulik, Pandharipande, and Thomas, and was shown to give rise to modular forms. In joint work with Benjamin Bakker we defined and computed analogous sheaf counts on surfaces via stable pairs of sheaves and showed that, in the case of K3 surfaces, the theory gives rise to modular forms of level 4.

Wednesday, April 2 and April 9, 2014, 3:00-4:00

Speaker

Claudia Polini (Notre Dame)

Title

Socles and Integrality

Abstract

The concepts of integral extensions, integral closures of rings, and integral dependence of ideals and modules are of central importance in commutative algebra. They play a crucial role in algebraic geometry, intersection theory, and the theory of multiplicities. In a series of papers with Corso, Huneke, Ulrich we compute part of the integral closure of an ideal and relate it to iterated socles (socles of socles) for which we give explicit formulas.

Wednesday, April 16, 2014, 3:00-4:00

Speaker

Juan Migliore (Notre Dame)

Title

Almost maximal growth of the Hilbert function

Abstract

Click here.