# Algebraic Geometry

### Varieties, Polyhedra, Computation

##### Winter Semester 2019/2020

# Abstracts of the Brill-Noether workshop

Below there are the abstracts of the talks of the Brill-Noether workshop (4th-6th November 2019) of the thematic semester Algebraic Geometry.

### Arend Bayer, University of Edinburgh

#### Brill-Noether statements in K3 categories

I will present a conjecture on Brill-Noether statements for stable objects in K3 categories, along with some evidence for the conjecture. Moreover, I will show how instances of the conjecture that can be proven yield an infinite sequence of Hassett-divisors in the moduli space of cubic fourfolds whose members can be characterised via the existence of a special surface on that cubic. .

### Dave Jensen, University of Kentucky

#### Classification of special divisors on chains of loops

In this talk, we will discuss the tropical approach to studying linear series on algebraic curves. As an application, we will see how combinatorial techniques can be used to prove the well-known Brill-Noether theorem of Griffiths and Harris.

#### Maximal rank for vertex avoiding divisors

In this talk we will discuss an approach to the maximal rank conjecture (now a theorem due to Eric Larson) and its variants using the technique of tropical independence. We will specifically focus on cases of the strong maximal rank conjecture for quadrics in genus 22 and 23, which are expected to have applications to the birational geometry of moduli spaces. We will describe the basic strategy used to prove this conjecture for a dense open subset of divisor classes — the so-called "vertex avoiding" divisor classes — in the relevant cases.

### Eric Larson, Stanford University.

#### The Maximal Rank Conjecture

Curves in projective space can be described in either parametric or Cartesian equations. We begin by describing the Maximal Rank Conjecture, formulated originally by Severi in 1915, which prescribes a relationship between the "shape" of the parametric and Cartesian equations --- that is, which gives the Hilbert function of a general curve of genus g, embedded in P^r via a general linear series of degree d. We then explain how recent results on the interpolation problem can be used to prove this conjecture.

### András Némethi

#### The Abel map associated with surface singularities

We fix a complex normal surface singularity \((X,o) \) whose link is a rational homology sphere. We also fix a good resolution \(\widetilde{X}\to X \) with exceptional curve \( E \) and a cohomology class \(l'\in H^1(\widetilde {X}, {\mathbb Z}) \). Then for any effective divisor \(Z \) supported on \(E \) we consider the affine space \({\rm Pic}^{l'}(Z) \) of line bundles over \({\mathcal O}_Z \) with Chern class \(l'\), and also the space of effective Cartier divisors \({\rm ECa}^{l'}(Z) \) over \(Z \) with Chern class \(l'\). \({\rm ECa}^{l'}(Z) \) is a smooth quasiprojective variety and the natural map \(c^{l'}(Z):{\rm ECa}^{l'}(Z)\to {\rm Pic}^{l'}(Z) \) is an algebraic morphism. In the talk we give an introduction to the main properties of \({\rm ECa}^{l'}(Z) \) and \(c^{l'}(Z) \) and we link them with the analytic and topological invariants of the singularity. The topological invariants are connected with the link, or with the intersection lattice and the Riemann-Roch expression. The analytic ones are connected with cohomology of line bundles on \(\widetilde{X}\). The Abel map depends essentially on the choice of the analytic structure on \((X,o)\) as well. We present several concrete formulae (e.g. the dimension of the image of the Abel map) in the case of specific analytic structures (weighted homogeneous, superisolated). In the case of generic analytic structure most of the invariants can be described topologically.

### Rahul Pandharipande, ETH Zürich

#### Abel-Jacobi maps and double ramification cycles

I will discuss several current developments in the study of Abel-Jacobi maps with connections to double ramification cycles, log geometry, and Gromov-Witten theory.

### Sam Payne, University of Texas at Austin

#### Tropical independence

In this talk, I will explain the method of tropical independence for verifying linear independence of sections of a line bundle on an algebraic curve. The method is based on the natural identification of skeletons of semistable models of the curve with spaces of valuations on its function field. I will also introduce an effective criterion for verifying tropical independence, and give simple examples (e.g., on the projective line) illustrating how tropical independence controls the combinatorial possibilities for the tropicalization of a linear series.

#### The non vertex avoiding case

In this talk I will discuss the additional constructions that are required to adapt the arguments from the vertex avoiding case to prove the strong maximal rank conjecture for quadrics in the cases (g,r,d) = (22,6,25) and (23,6,26). The key steps are: the definition of building blocks, the identification and classification of special pencils, and the construction and verification of the master template.

### John Sheridan, Stony Brook University

#### Continuous families of divisors on symmetric powers of curves

For X a smooth projective variety, we consider its set of effective divisors in a fixed cohomology class. This set naturally forms a projective scheme and if X is a curve, this scheme is a smooth, irreducible variety (fibered in linear systems over the Picard variety). However, when X is of higher dimension, this scheme can be singular and reducible. We study its structure explicitly when X is a symmetric power of a curve.

### Martin Ulirsch, Goethe-Universität Frankfurt

#### Tropical Prym-Brill-Noether theory:

Brill-Noether theory studies loci of special divisors in the Picard variety of an algebraic curve. We will be concerned with the study of such loci when the curve admits certain symmetries, in particular when it admits a fixed point free involution. In this case, Welters has introduced the study of so-called Prym-Brill-Noether loci in a torsor over the Prym variety of the associated unramified double cover. He, in particular, has shown a version of a Gieseker-Petri Theorem for these loci, which, together with an existence result due to Bertram, proves a natural analogue of the Brill-Noether Theorem for generic unramified double covers. In this talk I will introduce a new approach towards studying Prym-Brill-Noether loci using tropical methods. These methods allow us to give a new tropical proof of the Prym-Brill-Noether theorem independent of the one using Welters’ results. Moreover, our methods allow us to also consider the case of generic double covers where the base curve has a certain fixed gonality. In this case we obtain a new upper bound on the dimension of the Prym-Brill-Noether loci. The crucial technical ingredients are a new version of Pflueger’s decorated Young tableaux describing tropical divisor configurations on a so-called folded chain of loops. Given time, I will also outline a conjectural approach towards a lower bound on the dimension of these loci using methods from logarithmic Gromov-Witten theory. This is based on joint work with Yoav Len and (in parts) Dhruv Ranganathan.

### Filippo Viviani, Università Roma Tre

#### On the Universal Jacobian: algebraic, tropical and logarithmic aspects

I will start by reviewing how to compactify the universal Jacobian stack, parametrizing pointed curves endowed with a line bundle, over the moduli stack of stable pointed curves. I will introduce tropical universal Jacobians and describe the fibers of the forgetful morphism towards the moduli space of tropical curves. Then I will show that the tropical universal Jacobians are the skeletons of the Berkovich analytifications of the corresponding compactified universal Jacobian stacks. Finally, I will discuss logarithmic universal Jacobians and their tropicalization morphism towards the corresponding tropical universal Jacobians.