### Session S35 - Moduli Spaces in Algebraic Geometry and Applications

## Talks

Wednesday, July 14, 12:00 ~ 12:40 UTC-3

## Blown-up toric surfaces with non-polyhedral effective cone

### Ana-Maria Castravet

#### Université de Versailles, France - This email address is being protected from spambots. You need JavaScript enabled to view it.

We construct examples of projective toric surfaces whose blow-up at a general point has a non-polyhedral effective cone, both in characteristic 0 and in prime characteristic. As a consequence, we prove that the effective cone of the Grothendieck-Knudsen moduli space of stable, n-pointed, rational stable curves, is not polyhedral if n>=10 in characteristic 0 and in positive characteristic for an infinite set of primes of positive density.

Joint work with Antonio Laface (Universidad de Concepcion, Chile), Jenia Tevelev (University of Massachusetts at Amherst, USA) and Luca Ugaglia (Universita degli studi di Palermo, Italy).

Wednesday, July 14, 12:40 ~ 13:20 UTC-3

## Rationality of moduli spaces

### Inder Kaur

#### Pontifı́cia Universidade Católica do Rio de Janeiro (PUC-Rio), Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

In 2003, Larsen and Lunts established a connection between motivic integration and stable rationality. Recently, Nicaise and Shinder used this to prove that stable rationality is preserved under certain degeneration of smooth families. Theses ideas have been further developed by Kontsevich and Tschinkel. In this talk, I will survey these results and discuss how they can be applied to detecting rationality for certain moduli spaces. This is joint work in progress with Ananyo Dan.

Wednesday, July 14, 13:20 ~ 14:00 UTC-3

## Hyperkähler geometry of moduli of parabolic Higgs bundles in genus 0

### Claudio Meneses

#### Christian-Albrechts-Universität zu Kiel, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.

I will present a bird's-eye view of work in progress whose aim is to understand the hyperkähler structure of moduli spaces of stable parabolic Higgs bundles, as explicitly as possible, in the case of genus 0. This problem is intimately related to certain algebro-geometric structures of the moduli space, and is particularly sensitive to variations of parabolic weights and wall-crossing phenomena.

Wednesday, July 14, 14:00 ~ 14:40 UTC-3

## Moduli varieties of twisted local systems

### Florent Schaffhauser

#### Universidad de Los Andes, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.

Complex analytic orbi-curves give rise to natural examples of twisted local systems, for which the fundamental group acts non-trivially on the coefficients. In this talk, we construct moduli varieties of twisted local systems, and prove that these are affine varieties over the complex numbers, whose (strong) topology can be studied through an appropriate generalization of the non-Abelian Hodge correspondence to the case of nonconstant coefficients. This partially answers a question of Carlos Simpson on the meaning of the Dolbeault moduli space in the nonconstant case.

Wednesday, July 14, 14:40 ~ 15:20 UTC-3

## Brill--Noether theory over the Hurwitz space

### Isabel Vogt

#### University of Washington/Brown University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let C be a curve of genus g. A fundamental problem in the theory of algebraic curves is to understand maps of C to projective space of dimension r of degree d. When the curve C is general, the moduli space of such maps is well-understood by the main theorems of Brill--Noether theory. However, in nature, curves C are often encountered already equipped with a map to some projective space, which may force them to be special in moduli. The simplest case is when C is general among curves of fixed gonality. Despite much study over the past three decades, a similarly complete picture has proved elusive in this case. In this talk, I will discuss joint work with Eric Larson and Hannah Larson that completes such a picture, by proving analogs of all of the main theorems of Brill--Noether theory in this setting.

Joint work with Eric Larson (University of Washington/Brown University) and Hannah Larson (Stanford University).

Wednesday, July 14, 15:20 ~ 16:00 UTC-3

## The Kodaira dimension of the moduli space of curves: recent progress on a century old problem.

### Gavril Farkas

#### Humboldt Universität Berlin, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.

The problem of determining the birational nature of the moduli space of curves of genus g has received constant attention in the last century and inspired a lot of development in moduli theory. I will discuss progress achieved in the last 12 months. In particular, making essential of tropical methods it has been showed that both moduli spaces of curves of genus 22 and 23 are of general type (joint with D. Jensen and S. Payne).

Thursday, July 15, 12:00 ~ 12:40 UTC-3

## Moduli spaces of unstable sheaves via non-reductive GIT

### Victoria Hoskins

#### Radboud University Nijmegen, The Netherlands - This email address is being protected from spambots. You need JavaScript enabled to view it.

Reductive GIT has enabled the construction of many important moduli spaces and gives rise to natural notions of stability. However, non-reductive groups naturally appear in certain moduli problems (such as vector bundles of fixed Harder-Narasimhan type, where there is naturally a parabolic group action). In this talk, I will give an introduction to non-reductive GIT and explain how to take quotients by parabolic groups. Finally I will explain how this can be applied to construct moduli spaces of unstable objects (and in particular, moduli spaces of unstable sheaves).

Joint work with G. Berczi (Aarhus University), J. Jackson (Imperial College London) and F. Kirwan (University of Oxford).

Thursday, July 15, 12:40 ~ 13:20 UTC-3

## Open problems in geography of surfaces

### Giancarlo Urzúa

#### Pontificia Universidad Católica de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

I will survey on various aspects of geography of surfaces of general type (e.g. fixed fundamental group, rigid, hyperbolic, etc) and related open questions.

Thursday, July 15, 13:20 ~ 14:00 UTC-3

## Global Prym-Torelli theorem for ramified double coverings

### Angela Ortega

#### Humboldt Universität zu Berlin, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.

Given a finite morphism between smooth curves one can canonically associate it a polarised abelian variety, the Prym variety. This induces a map from the moduli space of coverings to the moduli space of polarised abelian varieties, known as the Prym map. It is a classical result that the Prym map is generically injective for étale double coverings.

In this talk we will give an introduction to the Prym varieties and the Prym maps. We will then consider the Prym map between the moduli space $ \mathcal{R}_{g,r}$ of double coverings over a genus g curve ramified at r points and $\mathcal{A}^{\delta}_{g-1+r/2} $ the moduli space of polarised abelian varieties of dimension $g-1+r/2$ with polarisation of type $\delta$. Unexpectedly, in the ramified case the Prym map is everywhere an embedding when $ r \geq 6$ and $g>0$. We will present a constructive proof of this result.

Joint work with Juan Carlos Naranjo (Universidad de Barcelona).

Thursday, July 15, 14:00 ~ 14:40 UTC-3

## Geometric stability conditions under autoequivalences

### Cristian Martinez

#### Universidade Estadual de Campinas , Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

There is a natural action of the group of autoequivalences of the derived category of a variety on its stability manifold. On a surface, most of the applications of the theory of Bridgeland stability conditions to the study of the geometry of moduli spaces of Gieseker semistable sheaves come from the study of the wall-crossing phenomena for families of geometric stability conditions (stability conditions for which all the skyscraper sheaves are stable). However, it is not always easy to identify nontrivial autoequivalences (other than shifting, and tensoring by line bundles), and even if we do, it is not always the case that the image of a geometric stability condition by such autoequivalence is again geometric. When this happens, we get a valuable tool to get results about the non-emptiness, projectivity, and even irreducibility of some Bridgeland moduli spaces. In this talk, I will present some instances where stability is preserved by a nontrivial autoequivalence, including elliptic and blow-up surfaces.

Joint work with Jason Lo (California State University, Northridge, USA).

Thursday, July 15, 14:40 ~ 15:20 UTC-3

## Special surfaces on special cubic fourfolds

### Emanuele Macri

#### Université Paris-Saclay, France - This email address is being protected from spambots. You need JavaScript enabled to view it.

I will report on a possible characterization of Hassett divisors on the moduli space of cubic fourfolds by the property of containing special surfaces. We will discuss a construction of such special surfaces for infinitely many divisors and the relation with the work of Russo and Staglianò on rationality of such cubics in low discriminant.

Joint work with Arend Bayer (University of Edinburgh), Aaron Bertram (University of Utah) and Alex Perry (University of Michigan).

Thursday, July 15, 15:20 ~ 15:40 UTC-3

## Moduli spaces of quiver representations and an open conjecture

### Kaveh Mousavand

#### Queen's University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

In 2014, Adachi-Iyama-Reiten introduced $\tau$-tilting theory of finite dimensional algebras as a modern generalization of tilting theory. The subject was primarily motivated by the notion of mutation in cluster algebras. It soon received a lot of attention and became an active area of research. Around the same time, Chindris-Kinser-Weyman studied the moduli spaces of quiver representations, in particular the behaviour of Schur representations of finite dimensional algebras. In this talk, we relate these two subjects. More specifically, we propose a conjectural geometric counterpart for the algebraic notion of $\tau$-tilting finiteness. For the sake of tractability, we avoid the technicality of $\tau$-tilting theory and state our conjecture as a modern analogue of the celebrated Brauer-Thrall conjectures. We verify the conjecture for some families of algebras and outline some strategies to treat the general case via a reductive argument.

Joint work with Charles Paquette (Royal Military College, Canada).

Thursday, July 15, 15:40 ~ 16:00 UTC-3

## On the Segre Invariant for Rank Two Vector Bundles on $\mathbb{P}^2$

### Leonardo Roa Leguizamón

#### Universidad Michoacana de San Nicolas de Hidalgo, Mexico - This email address is being protected from spambots. You need JavaScript enabled to view it.

We extend the concept of the Segre Invariant to vector bundles on a surface $X$. For a vector bundle $E$ of rank 2 on $X$, the Segre invariant is defined as the minimum of the differences between the slope of $E$ and the slope of all line subbundles of $E$. This invariant defines a semicontinuous function on the families of vector bundles on $X$. Thus, the Segre invariant gives a stratification of the moduli space $M_{X,H} (2; c_1, c_2)$ of $H$−stable vector bundles of rank 2 and fixed Chern classes $c_1$ and $c_2$ on the surface $X$ into locally closed subvarieties $M_{X,H} (2; c_1, c_2; s)$ according to the value of $s$. For $X=\mathbb{P}^2$ we determine what numbers can appear as the Segre Invariant of a rank $2$ vector bundle with given Chern's classes. The irreducibility of strata with fixed Segre invariant is proved and its dimensions are computed. Finally, we present applications to the Brill-Noether Theory for rank $2$ vector bundles on $\mathbb{P}^2.$ (see https://arxiv.org/pdf/2003.02727.pdf) .

Joint work with H. Torres-López and A.G. Zamora (Universidad Autónoma de Zacatecas, Mexico).

Tuesday, July 20, 16:00 ~ 16:40 UTC-3

## Gale Duality, Blowups and Moduli Spaces

### Carolina Araujo

#### IMPA, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

Gale correspondence provides a duality between sets of $n$ points in projective spaces $\mathbb{P}^s$ and $\mathbb{P}^r$ when $n=r+s+2$. For small values of $s$, this duality has a remarkable geometric manifestation: the blowup of $\mathbb{P}^r$ at $n$ points can be realized as a moduli space of vector bundles on the blowup of $\mathbb{P}^s$ at the Gale dual points. We explore this realization to describe the birational geometry of blowups of projective spaces at points in very general position.

Joint work with Ana-Maria Castravet (University of Versailles, France), Inder Kaur (PUC-Rio, Brazil) and Diletta Martinelli (University of Amsterdam, Netherlands).

Tuesday, July 20, 16:40 ~ 17:20 UTC-3

## Effective cone of the moduli space of sheaves on the plane via minimal free resolutions

### César Lozano Huerta

#### UNAM, Mexico - This email address is being protected from spambots. You need JavaScript enabled to view it.

The minimal free resolution of a sheaf is a complex whose factors are sums of line bundles. On the other hand, the generalized Gaeta resolution of a general sheaf on the plane is a complex whose factors are non-trivial vector bundles of higher rank.

In this talk I will discuss the relation between these two resolutions and a dictionary between them. This dictionary recovers the effective cone of the moduli space of sheaves on the plane, and based on it, we propose an elementary algorithm to compute the stable base locus decomposition of the Hilbert scheme of points on the plane. We prove that this algorithm is correct in some cases.

Tuesday, July 20, 17:20 ~ 18:00 UTC-3

## The Vafa-Witten equations and $T$-branes

### Ruxandra Moraru

#### University of Waterloo, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

The Vafa-Witten equations are a higher-dimensional analogue of the Hitchin equations on compact Riemann surfaces for oriented four-manifolds. On a compact complex surface, their solutions are polystable Higgs bundles (with Higgs fields taking values in a vector bundle twisted by the canonical bundle of the surface); T-branes are solutions whose Higgs fields are non-abelian. In this talk, we describe the geometry of T-branes and prove, in particular, that they can only exist on properly elliptic surfaces and surfaces of general type. We also give examples.

Joint work with Fernando Marchesano and Raffaele Savelli.

Tuesday, July 20, 18:00 ~ 18:40 UTC-3

## Variations of Hodge Structures of Rank Three $k$-Higgs Bundles and Moduli Spaces of Holomorphic Triples

### Ronald Zúñiga-Rojas

#### Universidad de Costa Rica, Costa Rica - This email address is being protected from spambots. You need JavaScript enabled to view it.

There is an isomorphism between the moduli spaces of $\sigma$-stable holomorphic triples and some of the critical submanifolds of the moduli space of $k$-Higgs bundles of rank three, whose elements $(E,\varphi^k)$ correspond to variations of Hodge structure, VHS. There are special embeddings on the moduli spaces of $k$-Higgs bundles of rank three. The main objective here is to talk about the cohomology of the critical submanifolds of such moduli spaces, extending those embeddings to moduli spaces of holomorphic triples.

Tuesday, July 20, 18:40 ~ 19:20 UTC-3

## Generalized hyperpolygons, meromorphic Higgs bundles, and integrability

### Steven Rayan

#### quanTA / University of Saskatchewan, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, I will construct a moduli space of generalized hyperpolygons from a comet-shaped quiver. The resulting Nakajima quiver variety can be interpreted as a distinguished subvariety of a moduli space of meromorphic Higgs bundles on a punctured curve. I will discuss how the moduli space inherits, for complete and minimal flags, a Gelfand-Tsetlin-type integrable system from the reduction of a product of cotangent bundles of (partial) flag varieties.

Joint work with Laura Schaposnik (University of Illinois at Chicago, USA).

Tuesday, July 20, 19:20 ~ 20:00 UTC-3

## Some Applications of Bridgeland Stability Conditions

### Aaron Bertram

#### University of Utah, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

A Bridgeland stability condition on a triangulated category can be thought of as a natural generalization of the Mumford slope stability of vector bundles on a Riemann surface. Inspired by mirror symmetry, stability conditions have been shown to have useful applications to problems in classical algebraic geometry. In this talk, we discuss two such applications: to the stability of restrictions of stable bundles to hypersurfaces (e.g. from a K3 surface to a curve on the surface), following work of Feyzbakhsh and to Torelli theorems for Kuznetsov kernels of certain Fano threefolds, following the work of several authors.