XXV CLA 2026

Tuesday, August 11

14:00-14:40 Puchuri

14:40-15:20 Moreira

15:20-16:00 Muñoz

Coffee break

16:20-16:45 Guajardo

16:45-17:10 Esquivel

17:10-17:35 Numpaque

Wednesday, August 12

 14:00-14:40 Troncoso

14:40-15:20 Rosas

15:20-16:00 Paucar

Jorge Esquivel Araya 

Gröbner bases and the moduli space of framed quivers

I will give a brief introduction to the basics of toric varieties, and how Gröbner bases can be used for compute them. I will illustrate how these tools can be applied to the moduli spaces of framed quivers. Finally, I present a recent method by Armenta et al. to do those computations.

Camila Guajardo López 

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 

Un algoritmo para encontrar cotas 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.

Gabriel Muñoz

Series lineales límite simples para curvas de tipo compacto

El objetivo de esta conferencia es presentar un estudio sobre series lineales límite simples, es decir, series lineales límite (en el sentido de Osserman) que tienen una base simple. Nos enfocaremos en el caso de curvas de tipo compacto con tres componentes irreducibles. Esta conferencia está basada en los artículos [1], [2] y [3] del expositor. El caso estudiado es bastante diferente del caso de dos componentes irreducibles. Por ejemplo, para curvas de tipo compacto con dos componentes irreducibles, toda serie lineal límite exacta es simple. Pero, para el caso de tres componentes irreducibles, esta propiedad ya no es satisfecha. Estudiaremos una cierta propiedad de distributividad relacionada con la existencia de una base simple y estableceremos caracterizaciones de dicha propiedad en cualquier multigrado fijo. Además, caracterizaremos a las series lineales límite exactas que son simples.

[1] MUÑOZ, G. Limit linear series for curves of compact type with three irreducible components. Communications in Algebra, 48:10, 4457-4482, 2020, DOI 10.1080/00927872.2020.176400.

[2] MUÑOZ, G. Abel Maps and Limit Linear Series for Curves of Compact Type with Three Irreducible Components. Bull Braz Math Soc. New Series, Volume 49, Issue 3, pp 549-575, 2018, DOI 10.1007/s00574-018-0071-2.

[3] MUÑOZ, G. Simple limit linear series for curves of compact

type with three irreducible components. ResearchGate, 2024, DOI  10.13140/RG.2.2.19160.51201.

Sebastián Numpaque

Tensor products of quiver bundles

In this work we introduce a notion of tensor product of (twisted) quiver representations with relations in the category of O_X-modules. As a first application of our notion, we see that tensor products of polystable quiver bundles are polystable and later we use this to both deduce a quiver version of the Segre embedding and to identify distinguished closed subschemes of GL(n,C)-character varieties of free abelian groups.

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.

Liliana Puchuri 

Negative curves and elliptic fibrations on a special rational surface

The complex projective plane blown up at nine of the twelve triple points of the dual Hesse arrangement carries infinitely many irreducible rational curves of self-intersection minus one, the so-called minus-one curves. Following Dumnicki and collaborators, these curves can be tracked through a diagram of bifurcations, obtained by repeatedly applying to a line the three quadratic Cremona maps that preserve the arrangement. The symmetric pattern of the degrees appearing in the diagram led them to ask for a closed formula for the degree of the rational curve at each entry. In this talk we solve this problem for an infinite diagram of bifurcations, and also describe the positions and multiplicities of the singularities of every rational plane curve in the diagram, together with their defining equations, computed in Python and in Singular. 

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.