The Berkeley-Stanford Algebraic Geometry Colloquium covers the full range of topics in algebraic geometry, and is intended for a broader audience than a typical research seminar. Graduate students and researchers in nearby fields are particularly welcome. Each meeting features two speakers chosen for their contributions to the field and their expository ability. There will be a dinner afterwards.

Organizers: David Eisenbud, Dagan Karp, Martin Olsson, and Brian Osserman (Berkeley) and Jun Li, Sam Payne, and Ravi Vakil (Stanford).

Tuesday October 3 meeting (in Berkeley): (poster)

Joe Harris (Harvard): Recent progress in Castelnuovo theory (3:45-4:45 pm, 2 Le Conte)

Over a century ago, Castelnuovo, Fano and others initiated a study of the geometry of curves of high genus in projective space; in particular, they established the link between the geometry of these curves and the geometry of configurations of points with small Hilbert function. Their results were extended in a series of conjectures by Eisenbud and Harris in the 1980s, which would give in principle a complete description of such curves.

Recently, Ivan Petrakiev has made the first substantial progress of these conjectures since their formulation. In this talk, I would like to review the context and significance of these conjectures, ending with a description of Petrakiev's work.

János Kollár (Princeton): Holonomy groups of algebraic vector bundles (5:00-6:00 pm, 2 Le Conte)

In this lecture I define the algebraic holonomy group of a stable vector bundle and compare it with the differential geometric holonomy group and the minimal structure group. We then start to study the interaction between the holonomy group and geometric properties of the bundle. (Joint work with Balaji, Biswas, Guralnick and Tiep)

Tuesday November 28 meeting (in Stanford): (poster)

Chuck Weibel (Rutgers): Non-smooth schemes, K-theory of R[t], and cdh cohomology (3:45-4:45, 383-N)

Voevodsky's cdh cohomology turns out to be the right tool for computing the K-theory of R[t] and relating it to various Hochschild homology obstructions to the smoothness of R in characteristic 0. Replacing Spec(R) by projective varieties, these obstructions are just the ``singular'' parts of the Hodge structure on H*(X). Armed with these computations, we are able to settle several conjectures made in the 1970's. This is a report on joint work with Cortinas and Haesemeyer, and with Schlichting.

Sam Grushevsky (Princeton): Intersection numbers of divisors on $\overline{\mathcal A}_g$ (5:00-6:00, 383-N)

The talk will start by explaining how toroidal compactifications $\overline{\mathcal A}_g$ of the moduli spaces ${\mathcal A}_g$ of (principally polarized) abelian varieties are constructed. We will then study the intersection numbers of divisors on a suitable toroidal compactification. It seems that most of these intersection numbers are zero, with only those essentially coming from top intersections on ${\mathcal A}_k$ for $k\le g$ being non-zero. We discuss the approaches to and partial results in proving this, computing the non-zero numbers, and generalizing to other symmetric domains. This is joint work with Erdenberger and Hulek.

Tuesday January 16 meeting (at MSRI): (poster)

Mark Kisin (Chicago): Modularity of 2-dimensional Galois representations (4:00-5:00, Simons Auditorium)

I will report on some of the recent advances towards the proof of Serre's conjecture, and explain the applications to modularity of Abelian varieties of GL_2 type, to Artin's conjecture and to 2-dimensional p-adic Galois representations. If time permits I will give some of the geometric ideas which go into the analysis of the deformation rings of local Galois representations. No prior knowledge about Serre's conjecture or modular forms will be assumed.

Christopher Hacon (Utah): Finite generation of pluricanonical rings (5:15-6:15, Simons Auditorium)

In this talk we will present recent progress in the minimal model program which has led to a proof of the following: Theorem: Let X be a smooth complex projective variety, then the pluricanonical ring R(K_X)=\oplus _{m\geq 0}H^0(X, mK_X) is finitely generated.

Friday April 6 meeting (in Stanford): (poster)

Phillip Griffiths (IAS): Algebraic Cycles and Singularities of Normal Functions (3:30-4:30, 383-N)

Normal functions provide a classical method for studying algebraic cycles and Hodge classes inductively by codimension. Previously the theory has been restricted to one dimensional base spaces on to higher dimensional families with smooth fibres. Moreover, the theory has been primarily useful to prove non-existence results, as the failure in general of Jacobi inversion makes the classical existence approach of Poincare-Lefschetz of limited value.

If, however, one goes to higher dimensional families with arbitrary singular fibres, then there is a rich interplay between algebraic cycles and the singularities of the associated normal function. For example the Hodge conjecture is equivalent to the existence of singularities of the associated normal function where the parameter space is the hypersurface sections of sufficiently higher degree. Moreover, the Hodge conjecture has certain algebro-geometric consequences for which no algebro-geometric argument is yet known. This talk will attempt to present an overview of the above. (This is a report on joint work with Mark Green.)

Luc Illusie (Paris-Sud): On Gabber's finiteness theorem in etale cohomology (5:00-6:00, 383-N)

Gabber has recently proven the constructibility of direct images of constructible sheaves of torsion prime to the characteristics by morphisms of finite type between excellent schemes. The main new ingredient in the proof is a deep local resolution theorem. I will start with a brief historical sketch, give an outline of the proof and discuss some by-products and related questions.

For previous joint seminars, click here.