WAGS / Fall 2006

Titles and Abstracts

University of Utah
11-12 November 2006

Speaker: Patrick Brosnan
Title: The essential dimension of an algebraic stack
Abstract: The essential dimension of algbebraic group is a numerical invariant invented by Buhler and Reichstein to measure the complexity of torsors for that group. In this lecture, I will report on joint work with Reichstein and Vistoli generalizing this notion to algebraic stacks.
Speaker: Paolo Cascini
Title: Minimal Model Program for Varieties of General Type
Abstract: TBA
Speaker: Tommaso de Fernex
Title: Birational rigidity of hypersurfaces
Abstract: Pukhlikov conjectured that for N greater or equal to 4, all smooth hypersurfaces of degree N in the N-dimensional projective space nonrational in a very strong sense, namely, that they are birationally superrigid, the case N=4 of this result being the celebrated theorem of Iskovskikh and Manin. In this talk I will give a proof of Pukhlikov's conjecture. After introducing the notion of birational rigidity and motivating it from the minimal model program, I will overview the methods (old and new) that are involved in the proof of the conjecture. This will lead me to talk about multiplier ideals, and to explan how arc spaces can be useful in order to study suitable restriction properties of these ideals.
Speaker: Paul Hacking
Title: Smoothable del Pezzo surfaces with quotient singularities
Abstract: We describe the classification of del Pezzo surfaces of Picard rank one with smoothable quotient singularities. These include the well known examples with A,D,E singularities, but there are many more where the canonical divisor is not Cartier. There are several infinite series of toric examples with a suprising combinatorial structure, and all but a finite number of exceptional cases are obtained as deformations of the toric examples if K^2 > 1. Joint work with Yuri Prokhorov.
Speaker: János Kollár
Title: Some interesting hypersurfaces in weighted projective spaces
Abstract: TBA
Speaker: Max Lieblich
Title: Boundedness of families of canonically polarized manifolds
Abstract: In 1962, Shafarevich conjectured (among other things) that the number of non-isotrivial families of proper smooth curves of genus g over a fixed smooth base curve B is finite. This conjecture, which was proven by Parshin (for proper B) and Arakelov (in general), has strong ties to arithmetic (Mordell's conjecture) and to the geometry of the moduli space of curves of genus g. I will describe joint work with S\'andor Kov\'acs in which we prove a generalization of this conjecture for families with higher-dimensional fibers over bases of arbitrary dimension. The proof is independent of the Minimal Model Program (although this is now perhaps irrelevant!).
Speaker: Sam Payne
Title: Toward foundations of tropical geometry
Abstract: Tropicalization, the degeneration of algebraic or analytic spaces to piecewise linear objects, has recently emerged as a powerful technique with applications to real and complex enumerative geometry, topology of real algebraic varieties, and the arithmetic of heights. In this talk, I will propose a definition of abstract tropical varieties as thickenings of polyhedral complexes with integral structure, building upon earlier ideas of Kempf, Knudsen, Mumford, and Saint-Donat, as well as recent work of Kontsevich, Mikhalkin, Soibelman, and many others.

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