UM/UIC/OSU Weekend Algebraic Geometry WorkshopMarch 1011, 2012, Ohio State University
Location: CH 240


TIME  SPEAKER  TITLE 
March 10, 2012
11:30am12:30pm  David Morrison
(UCSB) 
Elliptic fibrations revisited
(pretalk for graduate students only, 10:45am  11:15am) 
March 10, 2012
12:30pm1:45pm 
Lunch Break  
March 10, 2012
2:303:30pm 
Bhargav Bhatt
(Michigan) 
Comparison theorems in padic Hodge theory
(pretalk for graduate students only, 1:45pm  2:15pm) 
March 10, 2012
5:00pm6:00pm 
Izzet Coskun
(UIC) 
The birational geometry of the Hilbert scheme of points on P^2 and Bridgeland Stability
(pretalk for graduate students only, 4:15pm 4:45pm) 
March 11, 2012
10:30am11:30am 
Emanuele Macrì
(OSU) 
Line bundles on moduli spaces of complexes
(pretalk 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 Ftheory, and interest among physicists in Ftheory 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 padic 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 nonarithmeticallyinclined 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 padic schemes is canonically identified with classical crystalline cohomology. The talk will be introductory in nature: no background in padic 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 HsianHua Tseng
Does the page seem familiar?