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Seminari de Geometria Algebraica 2024/2025 |
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Conferenciant
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Títol |
Data i hora |
Robert Auffarth
Universidad de Chile
Contact:
jcnaranjo@ub.edu
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Counting principal polarizations on abelian varieties with automorphisms
In this talk we will explore the problem of counting the number of principal polarizations (modulo the action of the automorphism group) on a given abelian variety. More specifically, given a positive-dimensional irreducible component of the singular locus of the moduli space of principally polarized abelian varieties, we will want to answer the question of how many principal polarizations a very general member can have. We will give some motivation to this problem, and show that the answer to the previous question is not always 1!
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Divendres 4 d'octubre, 15h10, Aula T2, FMI-UB.
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Enric Florit Zacarias
Universitat de Barcelona
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Abelian varieties that split modulo all but finitely many primes
Let k be a number field and let A be an abelian variety defined over k.
We say A splits if it is isogenous to a product of abelian varieties of smaller dimension. Otherwise, A is simple. When A is simple, it may well happen that A splits modulo some prime p of k. We then have the following problem: given a simple abelian variety A, describe the set of primes p such that A mod p splits.
A standard fact says that the splitting behaviour of A is given by the semisimple decomposition of its endomorphism algebra. On the other hand, the endomorphism algebra of A modulo p was completely described by Honda and Tate by looking at the Frobenius endomorphism. Since End(A) is a subalgebra of End(A mod p), we can study the relation of one algebra to another.
In this talk we will explain the characterisation of subalgebras of division algebras. As a corollary we can show that, when End(A) is noncommutative, A splits modulo all but finitely many primes p of k.
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Divendres 18 d'octubre, 15h10, Aula T2, FMI-UB.
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Anatoli Shatsila
Jagiellonian University
Contact:
jcnaranjo@ub.edu
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Hyperelliptic genus 3 curves with involutions and Prym map
We will call a Galois covering of smooth curves Klein hyperelliptic if both curves are hyperelliptic and the Galois group is isomorphic to the Klein four-group. Borowka and Ortega (2023) proved that the Prym map for such coverings is injective. However, if one allows the bottom curve to have genus less than 2 then there is one additional non-trivial case, namely, covers f: C --> E with g(C) = 3 and g(E) = 1. I will show that the Prym map in this case is generically 2-to-1. I will also give a description of the moduli space of such covers by tuples of points on the projective line and use it to construct an involution on the moduli space induced by the Prym map. This is a joint work with Pawel Borowka.
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Divendres 25 d'octubre, 15h10, Aula T2, FMI-UB.
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Bernard Teissier
IMJ-PRG
Contact:
ana.belen.de.felipe@upc.edu
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A path to resolution of singularities through toric geometry
Resolution of singularities of a toric variety by toric birational maps is blind to the characteristic. I will explain a conjecture and some encouraging results about the use of this to resolve singularities
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Divendres 8 de novembre, 15h10, Aula T2, FMI-UB.
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Marc Masdeu
Universitat Autònoma de Barcelona
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Efficiently computing Theta functions for p-adic Schottky groups
Let K / Qp be a finite extension of the p-adics. A subgroup G
in GL2(K) is Schottky if it is finitely generated by hyperbolic
matrices. These groups are always discrete and free, and act on the
projective line minus a set of “bad points”. The quotient by this action
is known as a Mumford curve, and the p-adic theta functions allow for
the computation of their Jacobians, among other applications. In this
talk we will explain join work with Xavier Xarles, whereby we devise a
practical polynomial-time algorithm to compute these p-adic Theta functions.
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Divendres 15 de novembre, 15h10, Aula T2, FMI-UB.
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Marta Casanellas
UPC
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Obtaining equations for equivariant evolutionary models
Phylogenetics studies the evolutionary relationships among species using their molecular sequences. These relationships are represented on a phylogenetic tree or network. Modeling nucleotide or amino acid substitution along a phylogenetic tree is one of the most common approaches in phylogenetic reconstruction. One can use a general Markov model or one of its submodels given by certain substitution symmetries. If these symmetries are governed by the action of a permutation group G on the rows and columns of a transition matrix, we speak of G-equivariant models. A Markov process on a phylogenetic tree or network parametrizes a dense subset of an algebraic variety, the so-called phylogenetic variety.
During the last decade algebraic geometry has been used in phylogenetics for reconstructing phylogenetic trees and for establishing the identifiability of parameters of complex evolutionary models (and thus guaranteeing model consistency). Since G-equivariant models have fewer parameters than a general Markov model, their phylogenetic varieties are defined by more equations and these are usually hard to find. We will see that we can easily derive equations for G-equivariant models from the equations of a phylogenetic variety evolving under a general Markov model. As a consequence, we will discuss the identifiability of networks evolving under G-equivariant models.
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Divendres 22 de novembre, 15h10, Aula T2, FMI-UB.
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Martí Lahoz
Universitat de Barcelona
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Prym semicanonical pencils
In the moduli space of double étale covers of curves of genus g > 1, the
locus of covers of curves with a semicanonical pencil (a
theta-characteristic with two sections) is formed by two irreducible
divisors distinguished by the parity of the dimension of a certain space
of sections. I will explain the behavior of the Prym map on each of
them, which is significantly different, and has a rich geometry in the
low genus cases. At the end I will focus on the odd divisor and on
genera 5 and 6, which are especially interesting. This is joint work in
collaboration with Joan Carles Naranjo, Andrés Rojas and Irene Spelta.
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Divendres 29 de novembre, 15h10, Aula T2, FMI-UB.
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Joana Cirici
Universitat de Barcelona
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Configuration spaces of algebraic varieties
I will use the theory of weights in étale cohomology to give a simple and conceptual proof of a theorem of Kriz stating that a rational model for the ordered configuration space of a smooth complex projective variety is given by the second page of the Leray spectral sequence for the obvious inclusion, along with its only non-trivial differential. Our proof builds on Totaro's study of this spectral sequence, combined with a basic observation related to the formality of filtered dg-algebras. An advantage of this proof is that it allows for a generalization to study the p-adic homotopy type of configuration spaces for certain algebraic varieties defined over finite fields. This is joint work in progress with Geoffroy Horel.
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Divendres 20 de desembre, 15h10, Aula T2, FMI-UB.
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Ricardo Menares
Pontificia Universidad Católica de Chile
Contact:
sombra@ub.edu
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Divendres 17 de gener, 15h10, per decidir, FMI-UB.
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Pietro Speziali
Universidade Estadual de Campinas
Contact:
jcnaranjo@ub.edu
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Divendres 24 de gener, 15h10, per decidir, FMI-UB.
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David Senovilla Sanz
Universidad de Cantabria
Contact:
maria.alberich@upc.edu
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Saito bases for branches with one Puiseux pair
Let $C$ be a plane curve in $(\mathbb C^2,\mathbf 0)$. According to K. Saito, the $\mathbb C\{x,y\}$-module $\Omega^1[C]$ of holomorphic 1-formas with $C$ invariant is free of rank 2. We call Saito basis of $C$ to any basis of $\Omega^1[C]$. In this talk we explain how, by means of Saito bases, we can obtain a series of analytic invariants of $C$. In particular, we see how to generalize the family of invariants defined by Y. Genzmer. Moreover, when $C$ is an irreducible branch with a Puiseux pair, we explain the underlying combinatorics of an algorithm to compute a Saito basis.
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Divendres 31 de gener, 15h10, per decidir, FMI-UB.
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A. Bayer
S. Canning
P. Frediani
V.G. Alonso
F. Gounelas
B. Klinger
M. Lelli-Chiesa
E. Macrì
R. Pardini
L. Pertusi
G.P. Pirola
T. Zhang
X. Zhao
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Birational Geometry: from moduli to geography
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5-7 de febrer, FMI-UB
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Dumitru Stamate
University of Bucharest
Contact:
marchesi@ub.edu
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Divendres 14 de febrer, 15h10, Aula T2, FMI-UB.
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Contact:
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Divendres 21 de febrer, 15h10, Aula T2, FMI-UB.
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Ignasi Mundet
Universitat de Barcelona
Contact:
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Divendres 28 de febrer, 15h10, Aula T2, FMI-UB.
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Contact:
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Divendres 7 de març, 15h10, Aula T2, FMI-UB.
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Semon Rezchikov
Princeton University
Contact:
eva.miranda@upc.edu
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Divendres 14 de març, 15h10, Aula T2, FMI-UB.
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Contact:
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Divendres 21 de març, 15h10, Aula T2, FMI-UB.
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Contact:
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Divendres 28 de març, 15h10, Aula T2, FMI-UB.
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Contact:
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Divendres 4 d'abril, 15h10, Aula T2, FMI-UB.
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