Berkeley-Stanford Joint Algebraic Geometry Seminar 2003-2004
This seminar will happen on Tuesdays, roughly monthly, alternating between Stanford and Berkeley. There will be two lectures, with a break in between, and plenty of time for informal discussions. There will also be a dinner afterwards (subsidized for students). Drives and parking will be organized. For more information, please contact David Eisenbud (email@example.com) or Tom Graber (firstname.lastname@example.org) at Berkeley, or Jun Li (email@example.com) or Ravi Vakil (firstname.lastname@example.org) at Stanford. Here is a map of Stanford's campus, with parking indicated.
Click here for the Stanford Algebraic Geometry Seminar, and here for the Berkeley Commutative Algebra and Algebraic Geometry Seminar.
Tuesday, October 21, at Stanford (Rm. 200-107, click here for a map, and here for the poster):
Allen Knutson (Berkeley): Simplicial complexes and B-B decompositions (3:15-4:15)
Abstract: For M a compact manifold with a Morse function (or better, M a projective variety with an algebraic circle action with isolated fixed points), we get a decomposition of M into Morse strata. When we're lucky, this is a stratification, partially ordering the strata; a famous example is M a flag manifold with the Bruhat stratification, and the order complex of this partial order turns out to be homeomorphic to a ball.
I'll explain the right definition to replace the order complex when (as is more usual) this decomposition is not a stratification, and how it's motivated by ``Samuel-Rees filtrations'' and their associated degenerations. The end result is a ``path model'' for the coordinate ring of M, specializing to the celebrated Littelmann path model in the flag manifold case, but independent of any representation theory.
Sándor Kovács (Washington): Recent advances in the Minimal Model Program, after Shokurov (4:45-5:45)
Abstract: One of the major discoveries of the last two decades in algebraic geometry is the realization that the theory of minimal models of surfaces can be generalized to higher dimensional varieties. The major initial architects of the resulting theory in the 1980s were Y. Kawamata, J. Kollár, S. Mori, M. Reid, and V. V. Shokurov. They have built a theory of minimal models that works in all dimensions except for one crucial step: the existence and termination of flips. Flips are birational operations that only appear in higher dimensions and their definition does not assure their existence. Nevertheless they are essential to obtaining mimimal models. It has proved extremely difficult to show the existence of flips. Mori proved their existence in dimension three, which earned him the Fields Medal in 1990, but there has been very little advance in dimensions four and higher for a long time. Recently Shokurov introduced revolutionary new ideas that immediately gave a more theoretical proof of the three-dimensional case and may lead to a complete solution to the problem. In this talk, the Minimal Model Program will be introduced, including key definitions, theorems, and procedures. Flips will be defined and their importance discussed. Time permitting, Shokurov's new ideas will be discussed and put into perspective with regard to the previous ideas of the theory.
Tuesday, November 11, at Berkeley (Rm. 939 Evans Hall, poster):
Daniel Allcock (UT Austin): Hyperbolic Geometry and Real Cubic Surfaces (3:45-4:45)
Abstract: Only 2 percent of real cubic surfaces have 27 real lines. This and many other things become visible when you realize the moduli space of real cubic surfaces as a quotient of real hyperbolic 4-space. (Joint work with J. Carlson and D. Toledo.)
David Eisenbud (Berkeley/MSRI): Algebraic Sets of Minimal Degree? (5:00-6:00)
Abstract: Algebraic varieties (that is, irreducible algebraic sets) of minimal degree were classified by Del Pezzo (for surfaces) and Bertini (in all dimensions.) They appear in many places in classical algebraic geometry -- and almost every week in this seminar, in the special case of the twisted cubic. For various reasons the notion of ``minimal degree'' is not a very sensible one for algebraic sets in general, but there are good geometric conditions that mean "minimal degree" in the irreducible case and generalize well. In recent work with Green, Hulek, and Popescu we've achieved a rather simple classification, from which a number of surprising geometric results flow. And of course there's a connection with syzygies and regularity. I'll these ideas, and how they fit together.
Tuesday, February 10, at Berkeley (Rm. 939 Evans Hall, poster):
Harm Derksen (Michigan): Castelnuovo-Mumford regularity of subspace arrangements and Invariant Theory (3:45-4:45)
Abstract: Jessica Sidman and the speaker recently solved Sturmfels' Conjecture that the Castelnuovo-Mumford regularity of the intersection of d homogeneous linear ideals is at most d. I will discuss this result and various generalizations. These results give another proof of the Noether Gap Conjecture in Invariant Theory. I will also discuss some results regarding minimal free resolutions of invariant rings.
Maryam Mirzakhani (Harvard/Clay): Simple geodesics on hyperbolic surfaces and intersection theory on the moduli space of curves (5-6)
Abstract: In this talk we discuss a formula for calculating the Weil-Petersson volume of the moduli space of bordered Riemann surfaces with geodesic boundary components. We use the result to derive a recursive formula for the intersection numbers of tautological classes on the moduli space of curves with marked points.
Tuesday, March 30, at Stanford (Rm. 383-N, poster)
Joe Harris (Harvard): The Enriques conjecture; or, How canonical is the canonical bundle? (3:15-4:15)
Abstract: The question we're dealing with is this: Is there any way of associating to each smooth curve C of genus g --- or at least to each C in an open subset of moduli --- a line bundle on C, other than by taking powers of the canonical bundle? (The answer, by the way, is no: the canonical bundle is truly canonical.) This question was posed almost a century ago; a bogus proof was given by Franchetta in the '40s (as a result of which the statement is usually called Franchetta's conjecture), and a correct proof was given in the '80s by Harer and Mestrano, based on a topological argument of Harer's.
In fact, the statement is immediately implied by a stronger conjecture made by Enriques decades earlier. Enriques claimed (or suggested; it's not always clear) an analogous statement for the Severi variety, namely that the only ways of choosing a line bundle on a general plane curve C of degree d and genus g are combinations of the canonical bundle K_C and the hyperplane bundle O_C(1).
In this talk I'll discuss a little of the history of the Enriques conjecture, and variants of it; but the main purpose of the talk will be to give a proof of the conjecture that Deepee Khosla and I found recently.
Mark de Cataldo (Stony Brook): The Hodge theory of algebraic maps (4:45-5:45)
Abstract: I will discuss joint work with Luca Migliorini on new structures on the cohomology of projective manifolds.