ICG-2023 > Higher Dimensional Algebraic Geometry

# SESSION

# Higher Dimensional Algebraic Geometry

**ABSTRACTS**

**Intersection of two quadrics and Lagrangian fibration**

**Arnaud Beauville, Université Côte d’Azur, France**

Let \(X\) be a n-dimensional (smooth) intersection of two quadrics, and let \(T^*X\) be its cotangent bundle. I will show that the algebra of symmetric tensors on \(X\) is a polynomial algebra in n variables. The corresponding map \(T^*X —> \mathbb C^n\) is a Lagrangian fibration, which admits an explicit geometric description; its general fiber is an open subset of an abelian variety, again with a precise geometric description.

This is joint work with A. Etesse, A. Höring, J. Liu, C. Voisin.

**Jacobians of hyperelliptic curves with involutions**

**Pawel Borowka, Jagiellonian University, Poland**

Hyperelliptic curves with involutions are special objects from the point of Prym theory. This is because the Prym variety of a hyperelliptic double covering is the image of Jacobian of another (complementary) hyperelliptic curve. Motivated by results on Klein coverings of genus 2 curves (i.e. unbranched Galois coverings with the Klein-four group of deck transformations), we plan to describe hyperelliptic (possibly branched) Klein coverings of any genera. The aim of the talk is to show that all the necessary information is encoded in the points of the projective line and to show the injectivity of any Prym map.

The talk is based on a joint work with A. Ortega and a joint work with A. Shatsila.

**Higher-dimensional Fano varieties**

**Joaquín Moraga, University of California, Los Angeles, USA**

Fano varieties and Calabi-Yau varieties are two of the three building blocks of algebraic varieties. In this talk, we will discuss some recent developments on higher-dimensional Fano varieties and log Calabi-Yau pairs. The central topic will be the study of an invariant, called complexity, which measures how far is a given algebraic variety from being a toric variety.

**Weil-Petersson volumes, stability conditions and wall-crossing**

**Scott Mullane, Melbourne University, Australia**

We consider Weil-Petersson volumes of the moduli spaces of conical hyperbolic surfaces. The moduli spaces are parametrised by their cone angles which naturally live inside Hassett’s space of stability conditions on pointed curves. The space of stability conditions decomposes into chambers separated by walls. Assigning to each chamber a polynomial corresponding to the Weil-Petersson volume of a moduli space of conical hyperbolic surfaces, we use algebraic geometry to compute wall-crossing polynomials relating the polynomials to each other, and to the volume of the maximal chamber, given by Mirzakhani’s polynomial.

**Cyclic covers of an algebraic curve from an adelic viewpoint**

**Francisco Plaza, Universidad de Salamanca, Spain**

The classification of branched covers of \(\mathbb P^{1}\) (or a general Riemann surface) ultimately rests on analytical and topological results involving Riemann’s Existence Theorem and the topological fundamental group. A long-standing question is whether there exists a completely algebraic solution to this problem. We achieve this for \(p\)-cyclic covers of algebraic curves, using a strategy which is in principle extensible at least to general finite abelian covers. Our method employs the geometric adele ring of the curve and allows us to recover various important known results without appealing to the theory of complex Riemann surfaces.

**An introduction to the homology of intersection spaces**

**Abraham Rojas, Universidade de São Paulo, Brazil**

The (reduced) Homology of Intersection Spaces (HI) is a homology theory for singular spaces, introduced by Banagl in [1]. It is an alternative to Intersection Homology (IH) to generalize Poincaré Duality and other properties satisfied by the ordinary homology of smooth projective varieties. HI has the advantage of having internal cup products, and many functors can be applied to the resulting Intersection Spaces, which leads to richer invariants. Also, it was proved in [2] that HI is preserved by nearby smooth deformations, for of isolated singularities of hypersurfaces. In this talk we give an overview of this theory, we comment on its connections with Mirror Symmetry and present some advances in the case of toric varieties.

[1] M. Banagl. Singular spaces and Generalized Poincaré Complexes. Electronic Research Announcemnts in Mathematical Sciences, AIMS (2009).

[2] M. Banagl, L. Maxim. Deformation of Singularities and the Homology of Intersection Spaces. Journal of Topology and Analysis (2011).

[3] S. Ghaed. Intersection spaces and Toric Varieties. PhD Thesis on Heildelberg University (2023).

**On the degree of irrationality of low genus K3 surfaces**

**Andrés Rojas, Humboldt University, Germany**

Given a projective variety, one can understand rational maps to a projective space of the same dimension in terms of the associated kernel bundle. In the case of K3 surfaces of Picard rank \(1\), this allows to prove that rational maps of degree at most d (induced by the primitive linear system) move in families. I will explain how, by combining vector bundle techniques with derived category methods, one can study and characterize in many cases rational maps of minimal degree for polarized \(K3\) surfaces of genus up to \(14\). This can be seen as a preliminary step towards computing the degree of irrationality of these surfaces.

This is based on joint work with Federico Moretti, as well as previous work by himself.

**Period matrices**

**Anita Rojas, Universidad de Chile, Chile**

Deep geometric information about a polarized abelian variety \(A\), is captured by its period matrix \(\Pi_A=(D\, Z_A)\). For instance, \(\Pi_A\) defines the relation between the real and the complex coordinate functions of its lattice and of its vector space respectively. As a consequence, period matrices are useful tools to describe interesting loci of moduli spaces of abelian varieties. However, it is a hard problem to find a period matrix for a particular polarized abelian variety.

In this talk we will present two methods to compute the period matrix of a polarized abelian variety, depending on the given geometric information about it. We will briefly discuss some open questions regarding the moduli space of abelian varieties, and the Torelli locus, that we intend to approach through the study of period matrices. Finally, we will show a couple of applications.

This is part of a joint work with Rubí E. Rodríguez.

**Periods of algebraic cycles and applications to Hodge loci**

**Roberto Villaflor, Pontificia Universidad Católica de Chile, Chile**

In this talk we will explain some methods for computing periods of algebraic cycles inside hypersurfaces of projective simplicial toric varieties, and show how this data can be used to study components of the Hodge loci. As one of the main ingredients we will introduce an Artinian Gorenstein ideal associated to each Hodge cycle. If time permits we will also discuss relations between this ideal and the (variational) Hodge conjecture.

Session Organizers

**Robert Auffarth**

Universidad de Chile, Chile

**Juan Carlos Naranjo**

University of Barcelona, Spain