October 6 Omprokash Das (UCLA) |
Birational geometry of surfaces and 3-folds over imperfect fields |
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) |
Hilbert schemes of points on surfaces and their tautological bundles |
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) |
Dominating varieties by liftable ones |
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. |
November 10 at 4pm in AP&M 6402 (pre-talk at 3:30pm) Kenneth Ascher (MIT) |
Compactifications of the moduli space of elliptic surfaces |
I will describe a class of modular compactifications of moduli spaces of elliptic surfaces. Time permitting, I will also discuss recent work towards connecting these compactifications with various existing compactifications of the moduli space of rational elliptic surfaces. This is joint work with Dori Bejleri. |
November 29 at 2pm in AP&M 6402 Georg Oberdieck (MIT) |
Holomorphic anomaly equation for elliptic fibrations and beyond |
Physics predicts that the Gromov-Witten theory of Calabi-Yau threefolds satisfies two fundamental properties: Finite generation and a holomorphic anomaly equation. I will explain a recent conjecture with Pixton that extends these conjectures to all elliptic fibrations, and indicate how to prove it in several basic cases. If time permits, we will also discuss holomorphic anomaly equations for hyper-Kaehler varieties. |
December 4 at 2pm in AP&M 6402 Philip Engel (Harvard University) |
Tilings and Hurwitz Theory |
Consider the tilings of an oriented surface by triangles, or squares, or hexagons, up to combinatorial equivalence. The combinatorial curvature of a vertex is 6, 4, or 3 minus the number of adjacent polygons, respectively. Tilings are naturally stratified into all such having the same set of non-zero curvatures. We outline a proof that for squares and hexagons, the generating function for the number of tilings in a fixed stratum lies in a ring of quasi-modular forms of specified level and weight. First, we rephrase the problem in terms of Hurwitz theory of an elliptic orbifold---a quotient of the plane by an orientation-preserving wallpaper group. In turn, we produce a formula for the number of tilings in terms of characters of the symmetric group. Generalizing techniques pioneered by Eskin and Okounkov, who studied the pillowcase orbifold, we express the generating function for a stratum in terms of the q-trace of an operator acting on Fock space. The key step is to compute the trace in a different basis to express it as an infinite product, and apply the Jacobi triple product formula to conclude quasi-modularity. |
December 8 Gregory Pearlstein (Texas A&M University) |
Torelli theorems for special Horikawa surfaces and special cubic 4-folds |
We will discuss recent work with Z. Zhang on Torelli theorems for bidouble covers of a smooth quintic curve and 2 lines in the plane, and cubic 4-folds arising from a cubic 3-fold and a hyperplane intersecting transversely in P^{4}.
The talk for graduate students will be, "Abelian Varieties and the Torelli Theorem". I will explain what an Abelian variety is, and discuss the Torelli theorem for curves. |
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. Contact Jonathan Conder at jconder@ucsd.edu about problems with the website or posters.