UM/UIC/OSU Weekend Algebraic Geometry WorkshopMarch 10--11, 2012, Ohio State University
Location: CH 240
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| TIME | SPEAKER | TITLE |
| March 10, 2012
11:30am-12:30pm | David Morrison
(UCSB) |
Elliptic fibrations revisited
(pre-talk for graduate students only, 10:45am - 11:15am) |
| March 10, 2012
12:30pm--1:45pm |
Lunch Break | |
| March 10, 2012
2:30--3:30pm |
Bhargav Bhatt
(Michigan) |
Comparison theorems in p-adic Hodge theory
(pre-talk for graduate students only, 1:45pm - 2:15pm) |
| March 10, 2012
5:00pm-6:00pm |
Izzet Coskun
(UIC) |
The birational geometry of the Hilbert scheme of points on P^2 and Bridgeland Stability
(pre-talk for graduate students only, 4:15pm -4:45pm) |
| March 11, 2012
10:30am-11:30am |
Emanuele Macrì
(OSU) |
Line bundles on moduli spaces of complexes
(pre-talk for graduate students only, 9:45am - 10:15am) |
(Morrison): Elliptic fibrations played an important role in the classification of algebraic varieties during the 1960's, 1970's, and 1980's but have not been studied as intensively since the advent of Mori theory. However, since 1996 they have had an interesting application in physics known as F-theory, and interest among physicists in F-theory has markedly increased in the past few years. In this talk, we will discuss what Mori theory (and recent extensions of it) have to teach us about elliptic fibrations. We will also describe some new things we are learning about elliptic fibrations thanks to the interaction with theoretical physics.
(Bhatt): A fundamental theorem in classical Hodge theory is the isomorphism between de Rham and Betti cohomology for complex manifolds; this follows directly from the Poincare lemma. The p-adic analogue of this comparison lies much deeper, and was the subject of a series of conjectures made by Fontaine in the early 80s (following remarks of Grothendieck). In the last three decades, these conjectures have been proven by various mathematicians, and have had an enormous influence on modern arithmetic and algebraic geometry. In my talk, I will first discuss Fontaine's conjectures, and why non-arithmetically-inclined algebraic geometers might care about them. Then I will talk about some work in progress that leads to a new conceptual and relatively simple proof of these conjectures based on general principles in *derived* algebraic geometry (specifically, derived de Rham cohomology), and some classical geometry with curve fibrations.
The work presented builds on ideas of Beilinson who proved the de Rham comparison conjecture this way. The key new result that allows us to extend Beilinson's work to the crystalline/semistable setting is purely geometric: we show that derived de Rham cohomology for lci morphisms of p-adic schemes is canonically identified with classical crystalline cohomology. The talk will be introductory in nature: no background in p-adic Hodge theory or derived algebraic geometry will be needed.(Coskun): In this talk, I will describe joint work with Daniele Arcara, Aaron Bertram and Jack Huizenga on the birational geometry of the Hilbert scheme of points. We study the stable base locus decomposition of the effective cone of the Hilbert scheme. We describe many of the models and give them modular interpretations in terms of moduli spaces of Bridgeland stable objects.
(Macrì): One of the main problems with moduli spaces of stable complexes (with respect to stable sheaves) is that it is difficult to apply GIT techniques in their construction. As a consequence, many basic geometric questions regarding them are still open. For example, it is not known if they are all projective varieties.
In this talk, I will report on joint work in progress with A. Bayer on constructing positive line bundles on moduli spaces of complexes. We will construct, in general, a nef line bundle naturally associated to a stability condition and prove it is ample in the case of K3 surfaces (for a generic stability condition).This page is maintained by Hsian-Hua Tseng
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