Algebraic Geometry Seminar
Department of Mathematics
Princeton University
Spring 2008 Lectures
Regular meeting time:
Tuesdays 4:30-5:30 (Tea served at 3:30)
Place: Fine 322
| Date | Speaker | Title |
| Monday Feb 4 4:30 | Victor Przyjalkowski Steklov Institute, Moscow |
Weak Landau--Ginzburg models of Fano varieties
Mirror symmetry conjectures states that for any smooth Fano variety there exists a dual Landau--Ginzburg model, that is, a pencil of algebraic varieties whose symplectic properties transforms to algebraic ones of the initial Fano, and vice versa, whose algebraic properties transforms to symplectic ones. One of the main problems of mirror symmetry is finding of such models. We discuss a purely computational method of finding of natural candidates for dual models (which are called weak Landau--Ginzburg models). Finally we discuss their properties and their relations with toric degenerations. |
| Feb 5 | Dragos Oprea Stanford University |
Theta dualities on moduli spaces of sheaves
The moduli spaces of sheaves carry natural theta divisors, which are higher-rank analogues of the classical theta divisors on Jacobians. I will explain the general setting of Beauville/Le Potier's strange duality conjecture which relates sections of theta line bundles on two complementary moduli spaces. In particular, I will discuss numerical evidence in the case of sheaves over curves, K3 and abelian surfaces. This is joint work with Alina Marian. |
| Feb 12 | Ravi Vakil Stanford University |
Reimagining universal covers and fundamental groups in algebraic
and arithmetic geometry
In topology, the notions of the fundamental group and the universal cover are inextricably intertwined. In algebraic geometry, the traditional development of the étale fundamental group is somewhat different, reflecting the perceived lack of a good universal cover. However, I will describe how the usual notions from topology carry over directly to the algebraic and arithmetic setting without change, rectifying imperfections in the étale fundamental group. One key example is the absolute Galois group scheme, which contains more information than the traditional absolute Galois group, in a choice-free manner, and has a rich arithmetic structure. Its geometric fiber is the classical absolute Galois group as a topological group (the profinite topology is the Zariski topology, and comes from geometry). I will also discuss the example of abelian varieties and the Tate module. This is joint work with Kirsten Wickelgren. |
| Feb 19 | Max Lieblich Princeton University |
Moduli of orbifold twisted sheaves and the period-index
problem
|
| Friday Feb 29 |
Mikhail Kapranov (Yale) Kiran Kedlaya (MIT) James McKernan (MIT) | Columbia-Courant-Princeton algebraic geometry seminar at Columbia |
| Mar 4 | Klaus Hulek Leibniz Universität Hannover |
Moduli of polarized symplectic manifolds
In many ways irreducible symplectic manifolds behave similar to K3-surfaces, although it is known that the global Torelli theorem fails in general. Nevertheless, it is possible to relate moduli spaces of polarized irreducible symplectic manifolds to quotients of type IV domains by an arithmetic group. We will give an introduction to the subject and sketch a proof that certain moduli spaces of polarized irreducible symplectic manifolds are of general type. This is joint work with V. Gritsenko and G.K. Sankaran. |
| Monday Mar 10 4:30 | Gavril Farkas Humboldt Universität zu Berlin |
Maps between moduli spaces of curves and Gieseker-Petri
divisors
We study contractions of the moduli space of stable curves beyond the minimal model of M_g by resolving and giving a complete enumerative description of the rational map between moduli spaces of curves Mg --> Mh which associates to a curve C of genus g, the Brill-Noether locus of special divisors in the case this locus is a curve. As an application we construct myriads of moving effective divisors on M_g of small slope. For low g, our calculation can be used to study the intersection theory of the moduli space of Prym varieties of dimension 5. |
| Mar 18 | Spring break |
No seminar
|
| Mar 25 | Izzet Coskun University of Illinois at Chicago |
The birational geometry of Kontsevich moduli spaces
I will describe the stable base loci of linear systems on the Kontsevich moduli spaces of maps to projective spaces and Grassmannians. This description allows us to run the log minimal model program for these moduli spaces in small degree. I will give some examples where interesting classical moduli spaces occur. This is joint work with Dawei Chen and builds on previous work with Joe Harris and Jason Starr. |
| Apr 1 | Frédéric Mangolte
Université de Savoie |
Real singular Del Pezzo surfaces and rationally connected threefolds |