Alfonso Zamora Saiz

Universidad Politécnica de Madrid | UPM · Departamento de Matemática Aplicada (ETSI de Sistemas Informaticos)

Given a smooth projective variety X and a connected reductive group G defined over a field of characteristic 0, we define a moduli stack of principal $\rho$-sheaves that compactifies the stack of G-bundles on X. We apply the theory developed by Alper, Halpern-Leistner and Heinloth to construct a moduli space of Gieseker semistable principal $\rho$-...

This book introduces key topics on Geometric Invariant Theory, a technique to obtaining quotients in algebraic geometry with a good set of properties, through various examples. It starts from the classical Hilbert classification of binary forms, advancing to the construction of the moduli space of semistable holomorphic vector bundles, and to Hitch...

In this final chapter we review certain results on stratifications of the moduli space of Higgs bundles, performed with the invariants provided by the Harder-Narasimhan filtrations. The moduli space of Higgs bundles has two stratifications. The Shatz stratification arises from the Harder-Narasimhan type of the underlying vector bundle of the Higgs...

In this chapter we survey results on different correspondences of the stability notion and the GIT picture, at the level of maximal unstability provided by the Harder-Narasimhan filtration. In the main case of the moduli space of holomorphic vector bundles, we start with a stability notion for vector bundles in terms of their slope, for which a Har...

In this chapter, we collect all the necessary background to follow the further discussion on geometric invariant theory and moduli spaces. First we recover the notions of algebraic (affine and projective) variety and actions of algebraic groups, which will be the features in GIT quotients. Then we include a brief summary of sheaves, cohomology, and...

In this chapter, we recall Mumford’s construction of a moduli space of semistable holomorphic vector bundles over Riemann surfaces by using geometric invariant theory, and the notion of the Harder-Narasimhan filtration as the main tool to understand unstable bundles left out of the moduli space. We also give the basics of the analytical constructio...

The purpose of Geometric Invariant Theory (abbreviated GIT) is to provide a way to define a quotient of an algebraic variety X by the action of a reductive complex algebraic group G with an algebro-geometric structure. In this chapter we present a sketch of the treatment with a variety of examples. We also review the notion of stability from the di...

Let G be a complex reductive group and XrG denote the G-character variety of the free group of rank r. Using geometric methods, we prove that E(XrSLn) = E(XrPGLn), for any n,r ∈ N, where E(X) denotes the Serre (also known as E-) polynomial of the complex quasi-projective variety X, settling a conjecture of Lawton-Muñoz in [LM]. The proof involves t...

Presenting conclusions with the help of a graph can greatly improve your communication and convincing skills. R is a proficient tool for data visualization and in this chapter we explore some of the most well known plotting packages. First, with the R base graphics one can elaborate most of the fundamental graph styles with great level of customiza...

From Business Intelligence to advanced statistics applications, professionals are expected to access and manipulate large datasets, and R is the perfect tool for it. In this introductory chapter, we explain the principles of programming and the position of R in data science today. Then, a beginners level course on R starts introducing the main data...

The goal of data analysis is to use statistical tools to describe, infer, and predict values. The number of different concepts and techniques is very vast and it is essential to structure the way we learn them. Some descriptive statistics concepts are needed to understand the data we are working with. Then we can apply inference techniques to gener...

Prior to any data analysis, it is fundamental to be able to handle different sources and formats of information, such as files or web sites, and it is equally important to understand how to transform and manipulate all kinds of data so as to prepare everything in the right way to perform an statistical analysis. This chapter is divided into two par...

Let $\mathcal{X}_{\Gamma}G:=\mathrm{Hom}(\Gamma,G)/\!/G$ be the $G$-character variety of $\Gamma$, where $G$ is a complex reductive group and $\Gamma$ a finitely presented group. We introduce new techniques for computing Hodge-Deligne and Serre polynomials of $\mathcal{X}_{\Gamma}G$, and present some applications, focusing on the cases when $\Gamma...

This textbook offers an easy-to-follow, practical guide to modern data analysis using the programming language R. The chapters cover topics such as the fundamentals of programming in R, data collection and preprocessing, including web scraping, data visualization, and statistical methods, including multivariate analysis, and feature exercises at th...

Let $G$ be a complex reductive group and $\mathcal{X}_{r}G$ denote the $G$-character variety of the free group of rank $r$. Using geometric methods, we prove that $E(\mathcal{X}_{r}SL_{n})=E(\mathcal{X}_{r}PGL_{n})$, for any $n,r\in\mathbb{N}$, where E(X) denotes the Serre (also known as E-) polynomial of the complex quasi-projective variety $X$, s...

This survey intends to present the basic notions of Geometric Invariant Theory (GIT) through its paradigmatic application in the construction of the moduli space of holomorphic vector bundles. Special attention is paid to the notion of stability from different points of view and to the concept of maximal unstability, represented by the Harder-Naras...

With $G=GL(n,\mathbb{C})$, let $\mathcal{X}_{\Gamma}G$ be the $G$-character variety of a given finitely presented group $\Gamma$, and let $\mathcal{X}_{\Gamma}^{irr}G\subset\mathcal{X}_{\Gamma}G$ be the locus of irreducible representation conjugacy classes. We provide a concrete relation, in terms of plethystic functions, between the generating ser...

Given an infinite reductive algebraic group $G$, we consider $G$-equivariant coherent sheaves with prescribed multiplicities, called $(G,h)$-constellations, for which two notions of stability arise. The first one is analogous to the $\theta$-stability defined for quiver representations King and for $G$-constellations by Craw and Ishii, but dependin...

We give an example of an orthogonal bundle where the Harder-Narasimhan filtration, with respect to Gieseker semistability, of its underlying vector bundle does not correspond to any parabolic reduction of the orthogonal bundle. A similar example is given for the symplectic case.

This Ph.D. thesis studies the relation between the Harder-Narasimhan filtration and a notion of GIT maximal unstability. When constructing a moduli space by using Geometric Invariant Theory (GIT), a notion of GIT stability appears, which is determined by 1-parameter subgroups. This thesis shows a correspondence between the 1-parameter subgroup givi...

We prove that the filtration associated to the 1-parameter subgroup of Kempf giving the maximal way to destabilize, in the GIT sense, a point in the parameter space of the construction of the moduli space of rank 2 tensors over a smooth complex projective variety, does not depend on certain integer used in the construction of the moduli space, for...

We prove that the Harder-Narasimhan filtration for an unstable finite dimensional representation of a finite quiver coincides with the filtration associated to the 1-parameter subgroup of Kempf, which gives maximal unstability in the sense of Geometric Invariant Theory for the corresponding point in the parameter space where these objects are param...

An unstable torsion free sheaf on a smooth projective variety gives a GIT- unstable point in certain Quot scheme. To a GIT-unstable point, Kempf associates a "maximally destabilizing" 1-parameter subgroup, and this induces a filtration of the torsion free sheaf. We show that this coincides with the Harder-Narasimhan filtration. Then we prove the an...