October 6 Omprokash Das (UCLA)
Title: Birational geometry of surfaces and 3-folds over imperfect fields Abstract: Lots of progress have been made in the recent years on the birational geometry of surfaces and 3-folds in positive characteristic over algebraically closed field. The same can not be said about the varieties over imperfect fields. These varieties appear naturally in positive characteristic while studying fibrations (as a generic fiber). Recently the minimal model program (MMP) for surfaces over excellent base scheme was successfully carried out by Tanaka. He also showed that the abundance conjecture holds for surfaces over imperfect fields. His results have become one of main tools for studying fibrations in positive characteristic. One of the things that is not covered in Tanaka's papers is the del Pezzo surfaces (a regular surface with -K_X ample) over imperfect fields. One interesting feature of del Pezzo surfaces is that over an algebraically closed field they satisfy the Kodaira vanishing theorem. This makes the theory of del Pezzo surfaces quite interesting. However, over imperfect fields it was known for a while that in char 2, Kodaira vanishing fails for del Pezzo surfaces, due to (Schroer and Maddock). It is only very recently that some positive results started to show up. In a recent paper by Patakfalvi and Waldron it was shown that the Kodaira vanishing theorem holds for del Pezzo surfaces over imperfect fields in char p>3. In this talk I will show that in fact the Kawamata-Viehweg vanishing theorem holds for del Pezzo surfaces over imperfect fields in char p>3. I will also report on a project which is a work in progress (joint with Joe Waldron) on the minimal model program for 3-folds over imperfect fields and the BAB conjecture for del Pezzo surfaces over imperfect fields. |
October 13 David Stapleton (UCSD)
Title: Hilbert schemes of points on surfaces and their tautological bundles Abstract: Fogarty showed in the 1970s that the Hilbert scheme of n points on a smooth surface is itself smooth. Interest in these Hilbert schemes has grown since it has been shown they arise in hyperkahler geometry, geometric representation theory, and algebraic combinatorics. In this talk we will explore the geometry of certain tautological bundles on the Hilbert scheme of points. In particular we will show that these tautological bundles are (almost always) stable vector bundles. We will also show that each sufficiently positive vector bundle on a curve C is the pull back of a tautological bundle from an embedding of C into the Hilbert scheme of the projective plane. |
October 27 Remy van Dobben de Bruyn (Columbia University)
Title: Dominating varieties by liftable ones Abstract: Given a smooth projective variety over an algebraically closed field of positive characteristic, can we always dominate it by another smooth projective variety that lifts to characteristic 0? We give a negative answer to this question. |
December 8 Gregory Pearlstein (Texas A&M University)
Title: TBA Abstract: TBA |
Organizers: Elham Izadi, James McKernan and Dragos Oprea
This seminar is supported in part by grants from the NSF. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Past quarters: Fall 2013 , Winter 2014 , Spring 2014 , Fall 2014, The design of this webpage is copied shamelessly from the MIT Number Theory seminar site. Contact Jonathan Conder at jconder@ucsd.edu about problems with the website or posters.