XXV CLA 2026

Camila Guajardo Vásquez 

Un algoritmo para encontrar cotas sistólicas a partir de polígonos hiperbólicos

En geometría usualmente intentamos calcular invariantes de una subvariedad para caracterizar a la variedad ambiente que la contiene. Por ejemplo, entre las superficies de Riemann compactas de género dos, la superficie de Bolza se caracteriza por tener las sístoles más largas y el grupo de automorfismos más grande. ¿Están relacionados entonces el largo de las sístoles con las simetrías de una superficie?

Una sístole es una geodésica cerrada simple no homotópicamente trivial. La cantidad de ellas, hasta homotopía libre, se denomina kissing number. Este trabajo busca desarrollar un algoritmo para estimar tanto el largo de sístoles como el kissing number para superficies de género bajo. Para ello, el paso central es construir una representación geométrica de nuestra superficie S como dominio fundamental para la acción de un grupo Fuchsiano. 

A partir del polígono hiperbólico para S identificamos geodésicas cerradas que son candidatas a sístoles; sus largos dan una cota superior para el largo de las sístoles. La construcción incluye, de gratis, una acción por isometrías del grupo de automorfismos de S. Así, el tamaño de la órbita de una sístole da una cota inferior para el kissing number.

En esta presentación mostraré cómo funciona nuestro algoritmo en concreto y cuáles son las dificultades teórico-computacionales que aparecen al tratar con superficies de género superior.

Este trabajo fue parcialmente financiado por el proyecto Fondecyt Regular 1230034 “Subvarieties and loci in moduli spaces” (dirigido por la Prof. Anita Rojas Rodríguez), y por la Beca de Magíster Nacional 2025, folio 22250466, Subdirección de Capital Humano, ANID.

Miguel Moreira 

Rationality and symmetry of stable pairs generating series of Fano 3-folds

Stable pair invariants, also known as Pandharipande—Thomas invariants, are one of the various ways to "count" curves on 3-folds. A central conjecture in the field is that when these invariants are put together in a generating series, we obtain Laurent expansions of rational functions that are symmetric under the change of variable q->1/q. I will explain the proof (joint with I. Karpov) of this conjecture when the 3-fold is Fano, using new developments in wall-crossing formulas for moduli spaces of sheaves/complexes.

Rina Paucar Rojas 

Constant cycles curves and the Gysin kernel

In this talk I will talk about a result that relates the study of  constant cycles curves  (ccc), which are generalizations of rational curves, on surfaces and the study of the kernel of the Gysin homomorphism of Chow groups of 0-cycles induced by the embedding of a curve into a surface.

Alexander Quintero Vélez 

Moduli spaces of representations of superquivers

This talk presents ongoing work aimed at extending the framework introduced by Alastair King for moduli spaces of quiver representations to the setting of supergeometry. In the classical case, stability conditions allow one to construct well-behaved moduli spaces as geometric invariant theory quotients. Our goal is to develop an analogous theory for superquivers, naturally incorporating $\mathbb{Z}_2$-gradings and the language of superschemes.

I will introduce the notion of representations of superquivers and discuss possible formulations of stability in this context. Particular attention will be given to the challenges that arise when attempting to generalize classical constructions, including subtleties related to the geometry of superschemes and quotient constructions. As this work is still in progress, the emphasis will be on the conceptual framework.

Rudy Rosas 

An index theorem for holomorphic foliations on the projective plane

In this talk, we introduce an index associated with a foliation with respect to another foliation and a projective curve. The sum of these indices along the curve is an integer that depends only on the curve and the second foliation. When the curve is invariant under the first foliation, the classical Camacho-Sad index theorem is recovered.

Sergio Troncoso

Ulrich bundles on double and bidouble planes

Ulrich line bundles are known to be absent on general double covers of the projective plane by the work of Parameswaran and Narayanan (J. Algebra, 2021), while Sebastian and Tripathi (JPAA, 2022) established the existence of rank-two Ulrich bundles on general double planes. In this talk, I will present recent joint work with Juan Cruz Penagos and Jerson Caro showing that the same phenomenon persists for general bidouble covers. By determining the Picard number of these surfaces, we identify the exceptional situations in which Ulrich line bundles may exist. This provides the first systematic study of Ulrich bundles on non-cyclic abelian covers and highlights the rarity of rank-one Ulrich bundles in families of branched covers of the projective plane.