Schedule
Thursday 13th
14.00-14.50 Ciro Ciliberto Brill-Noether loci of rank two vector bundles on a general curve
15.00-15.50 John Christian Ottem Subvarieties with positivity properties
16.00-16.30 Coffee break
16.30-17.20 Jacopo Stoppa Stability data, connections, and curves
17.30-18.20 Diane Maclagan Tropical schemes, tropical cycles, and valuated matroids
The social dinner will take place from 20.15 at Antica Hostaria della Lanterna, via Giuseppe Mercalli 3, Milano.
Friday 14th
09.30-10.20 Lucia Caporaso Tropicalization of the moduli stack of algebraic curves
10.30-11.00 Coffee break
11.00-11.50 Martì Lahoz Products of theta divisors
12.00-12.50 Paolo Cascini On base point freeness in positive characteristic
Talks
Lucia Caporaso (Università degli Studi di Roma Tre)
Tropicalization of the moduli stack of algebraic curves
I will show how the moduli space of extended tropical curves can be identified with the “skeleton”, or “tropicalization”, of the classical moduli space of Deligne-Mumford stable curves. Joint work with Dan Abramovich (Brown University) and Sam Payne (Yale University).
Paolo Cascini (Imperial College London)
On base point freeness in positive characteristic
Many of the results in the Minimal Model Program depend on Kodaira vanishing theorem and its generalizations. On the other hand, because of the failure of these tools in positive characteristic, many of these results are still open in this case.
I will describe some recent progress towards the base point free theorem and the cone theorem over an algebraically closed field of positive characteristic.
Ciro Ciliberto (Università degli Studi di Roma Tor Vergata)
Brill-Noether loci of rank two vector bundles on a general curve
I will talk about work in collaboration with F. Flamini on Brill-Noether loci for rank two vector bundles on a curve curve with general moduli.
We describe the general member of some components of these loci as suitable extensions of line bundles.
The geometric counterpart of this is the analysis of minimal special curve sections of the corresponding projective bundles.
Martì Lahoz (Université Paris 7 Denis Diderot)
Products of theta divisors
I will present the following characterization of products of theta divisors: a subvariety of an abelian variety is a product of theta divisors if, and only if, it is normal and its desingularisation has holomorphic Euler characteristic 1. This is joint work with Zhi Jiang and Sofia Tirabassi.
Diane Maclagan (University of Warwick)
Tropical schemes, tropical cycles, and valuated matroids
The tropicalization of a subvariety of a torus is a set, together with some multiplicity data, that records the cycle of its compactification in an ambient toric variety. The set is encoded in the tropical scheme structure recently introduced by Jeff and Noah Giansiracusa.
In this talk I will introduce these notions, and outline how to also recover the tropical cycle (multiplicity data) from this information. The lurking combinatorics is that of valuated matroids. This is joint work with Felipe Rincon.
John Christian Ottem (University of Cambridge)
Subvarieties with positivity properties
A well-established principle in algebraic geometry is that geometric properties of an algebraic variety is reflected in the subvarieties which are in various senses ‘positively embedded’ in it. The primary example is the hyperplane section in a projective embedding of the variety, which gives rise to the notion of an ample divisor. However, in higher codimension it is less clear what it should mean in general for a subvariety to be ‘positive’.
We survey various definitions of positive embeddings and their geometric properties and discuss a related question of Peternell.
Jacopo Stoppa (Università degli Studi di Pavia)
Stability data, connections, and curves
During the last few years a number of results have appeared that link stability data on graded Lie algebras, Stokes factors for irregular connections, and tropical invariants. After a brief introduction to this circle of ideas, I will explain some new results obtained in collaboration with S. Filippini, M. Garcia-Fernandez, A. Mandini.
We construct a family of irregular meromorphic connections on P^1 which degenerates to the one introduced by Bridgeland and Toledano Laredo, and displays tropical behaviour in a different limit.