Algebraic Geometry Seminar

 Department of Mathematics
Princeton University

Fall 2006 Lectures

Regular meeting time: Tuesdays 4:30-5:30 (Tea served at 3:30)
Place: Fine 322

Date Speaker Title
Sep. 26 Nicholas Proudfoot
Columbia University 		Hypertoric varieties
A hypertoric variety is a quaternionic analogue of a toric variety. Just as the topology of toric varieties is closely related to the combinatorics of polytopes, the topology of hypertoric varieties interacts richly with the combinatorics of hyperplane arrangements and matroids. I will give an introduction to these spaces, and use arithmetic techniques to obtain combinatorial interpretations of the Betti numbers of hypertoric varieties, both for ordinary cohomology in the smooth case and intersection cohomology in the singular case.
Oct. 3 Dan Abramovich
Brown University		 Preconceptions and misconceptions on singularly relative stable maps
Joint work in progress with Barbara Fantechi, with closely related symplectic counterpart by Joshua Davis. A key tool in Gromov-Witten theory is the degeneration method, in which a variety is allowed to degenerate into two smooth components meeting transversally. Additional flexibility is obtained from more singular degenerations, which are the topic of this work. I'll review the old and discuss the new, emphasizing the surprises one meets on every turn.
Oct. 10
No seminar
Oct. 17 Sándor Kovács
University of Washington		 Generalizations of the Shafarevich Conjecture
At the 1962 ICM Shafarevich made a conjecture that predicted that for a fixed base and a fixed genus there are only finitely many non-isotrivial families of smooth projective curves of the given genus over the given base. This was proven in the function field case by Parshin (for a compact base) and by Arakelov (in general), and in the number field case by Faltings. In this talk I will discuss various higher dimensional generalizations of the function field case including Viehweg's conjecture and recent results in the area.
Oct. 24 Hershel Farkas
Hebrew University of Jerusalem; SUNY Stony Brook		 An elementary derivation of Thomae's formulae and some generalizations
In this talk I shall review the relevant facts concerning hyperelliptic Riemann surfaces and then show how the formulae of Thomae which connect the algebraic and transcendental parameters of a hyperelliptic surface can be derived as an almost immediate consequence of the definitions and elementary properties. More precisely, Thomae's formulae are polynomial equations relating theta constants and the branchpoints of the double cover of P^1 by the hyperelliptic curve. Time permitting, I shall say something about generalizations to other families of curves.
Oct. 31
Fall break - no seminar
Nov. 7 Aaron Bertram
University of Utah		 Constructing Bridgeland Moduli Spaces with Mukai Flops
Bridgeland stability for K3 surfaces, inspired by the string theorists' pi-stability, is based on a novel generalization of the "classical" slope function d/r for vector bundles on a curve. Unlike the curve case, Bridgeland stability has built-in variation and wall-crossings, leading one to predict new birational models of the moduli spaces of degree-stable coherent sheaves on K3 surfaces. In this talk I'll discuss a strategy worked out jointly with Daniele Arcara to exploit the holomorphic symplectic structure of these moduli spaces to construct the new birational models "by hand" as Mukai flops.
Nov. 14 Sam Payne
Stanford University; Clay Institute		 Toric vector bundles and the resolution property
Is every coherent sheaf on an algebraic variety the quotient of a locally free sheaf of finite rank? I will discuss an investigation of this question via equivariant vector bundles on toric varieties, and will give examples of complete (singular, nonprojective) toric threefolds with no nontrivial equivariant vector bundles of rank less than or equal to 3. It is not known whether these varieties have any nontrivial vector bundles at all.
Nov. 21 Mircea Mustaţă
University of Michigan; IAS 		Linear series, base loci and some convex geometry
Given a line bundle L on a complex variety X, one can define "moving intersection numbers" with every subvariety of X. When L admits a Zariski decomposition, i.e. when all sections of the multiples of L come from multiples of a positive part, these numbers are the usual intersection numbers with this positive part. I will discuss applications to the description of base loci, as well as connections with convex geometry.
Nov. 28 Brent Doran
Oxford University; IAS 		On some classical questions in affine geometry
Many well-known conjectures in affine geometry are, directly or indirectly, in essence questions about unipotent group actions. In another vein, many questions, especially about classification, have analogues in the topological world that are well-understood; A^1-homotopy theory provides a route to bring that topological reasoning into algebraic geometry. We focus mainly on the "simplest case" of principal G_a-bundle quotients and the A^1-homotopy type of a point, producing arbitrary dimensional families of non-isomorphic exotic affine spaces over a characteristic 0 field, as well as a conjecture for a general classification. In four real dimensions some of these may be algebraic structures on exotic smooth R^4's, although this seem difficult to check. Concrete examples will be given throughout.
Dec. 5 Günter Harder
Max Planck Institut für Mathematik; IAS 		Ordinary cohomology of arithmetic groups
Dec. 12 Brendan Hassett
Rice University 		Approximation for rationally connected varieties over function fields of curves
(joint with Yuri Tschinkel)
Let B be a smooth complex curve and X a variety smooth and proper over F=C(B). Graber/Harris/Starr have shown that if X is geometrically rationally connected then X(F) is nonempty. Building on work of Koll'ar/Miyaoka/Mori and others, we prove that X satisfies weak approximation at places of good reduction. There are partial results at places of bad reduction, e.g., for Del Pezzo surfaces and hypersurfaces of low degree. We also discuss new density results for integral points of log Fano pairs (X,D) over B.


Other seminars in this department

For more information about this seminar, contact Samuel Grushevsky and Max Lieblich.