# Seminars & Events for Algebraic Geometry Seminar

##### 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).

##### Noncommutative differential operators, unparametrized paths and Hodge structures

Joint Columbia-Courant-Princeton Algebraic Geometry Seminar at Columbia

##### The cone theorem revisited

Joint Columbia-Courant-Princeton Algebraic Geometry Seminar at Columbia

##### Measuring wild ramification using rigid geometry

Joint Columbia-Courant-Princeton Algebraic Geometry Seminar at Columbia

##### 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.

##### 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.

##### 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.

##### Real singular Del Pezzo surfaces and rationally connected threefolds

Recent results on classification of real algebraic threefolds will be described. Let W -> X be a real smooth projective threefold fibred by rational curves. J. Kollár proved that if the set of real points W(R) is orientable, then a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces. We proved sharp estimates on the number and the multiplicities of the Seifert fibres and on the number and the torsions of the lens spaces whenever X is a geometrically rational surface. These results answer in the affirmative three questions of Kollár. They are derived from a careful study of real singular Del Pezzo surfaces with only Du Val singularities. This is joint work with F. Catanese.

##### Wallcrossing for K-theoretic Donaldson invariants and computations for rational surfaces

Let $(X,H)$ be a polarized algebraic surface. Let $M=M^H_X(c_1,c_2)$ be the moduli spaces of $H$-semistable rank 2 sheaves on $X$ with Chern classes $c_1, c_2$. K-theoretic Donaldson invariants of $X$ are holomorphic Euler characteristics of determinant line bundles on $M$. These invariants are subject to wallcrossing when $H$ varies. In the first part of the lecture I present joint work with Nakajima and Yoshioka, where we determine a generating function for the wallcrossing in terms of elliptic functions. If time permits I will in the second part of the talk present some results about the K-theoretic invariants of rational surfaces, and relate these to Le Potiers strange duality conjecture.

##### Grothendieck duality via the homotopy category of flat modules

We will discuss a novel perspective of dualizing complexes which has been discovered in the last three years. We will review three recent articles, by Jorgensen, Krause and Iyengar-Krause, before coming to recent work by myself and by Murfet.