arXiv subjects

César Lozano Huerta

Publications and source records attributed to César Lozano Huerta.

8 recordsLinked to original sources

arXiv2026

Explicit birational geometry of determinantal quartic 3-folds

A general linear determinantal quartic in $\mathbb{P}^4$ is nodal, non-$\mathbb{Q}$-factorial and rational. We show that the family $\mathcal{F}$ of such quartics also contains rational $\mathbb{Q}$-factorial quartics, and that a generic member of $\mathcal{F}$ can specialize to a rational non-$\mathbb{Q}$-factorial double quadric. We prove that the birational geometry of these three types of 3-folds is governed by the extrinsic geometry of a curve $C\subset \mathbb{P}^3$ of degree 10 and genus 11.

math.AG↗

arXiv2024

The Noether-Lefschetz locus of surfaces in $\mathbb{P}^3$ formed by determinantal surfaces

We compute the dimension of certain components of the family of smooth determinantal degree $d$ surfaces in $\mathbb{P}^3$, and show that each of them is the closure of a component of the Noether-Lefschetz locus $NL(d)$. Our computations exhibit that smooth determinantal surfaces in $\mathbb{P}^3$ of degree 4 form a divisor in $|\mathcal{O}_{\mathbb{P}^3}(4)|$ with 5 irreducible components. We will compute the degrees of each of these components: $320,2508,136512,38475$ and $320112$.

math.AG↗

arXiv2021

Minimal free resolutions of sheaves on the projective plane and the stable base locus decomposition of their moduli spaces

The purpose of this paper is to incorporate minimal free resolutions into the study of the birational geometry of the moduli space of coherent sheaves on the plane with character $ξ$, denoted by $M(ξ)$. We show that it is possible to recover the relevant Bridgeland destabilizing object from the minimal free resolution in order to compute the effective cone $\mathrm{Eff}(M(ξ))$ and conjecture a full relationship between Bridgeland destabilizing objects and minimal free resolutions. Moreover, we also prove that minimal free resolutions, paired with interpolation for vector bundles, yield the movable cone of the Hilbert scheme of $n$ points on the plane $\mathbb{P}^{2[n]}$, for certain values of $n$. We propose a program that computes the full stable base locus decomposition of $\mathbb{P}^{2[n]}$ based on free resolutions and interpolation. We show that this programs yields correct answers for small values of $n$.

math.AG↗

arXiv2020

On the weak Lefschetz principle in birational geometry

This is an expository article written for the Notices of the AMS in which we discuss the weak Lefschetz Principle in birational geometry. Our departing point is the influential work of Solomon Lefschetz started in 1924. Indeed, we look at the original formulation of the Lefschetz hyperplane theorem in algebraic topology and build up to recent developments of it in birational geometry. In doing so, the main theme of the article is the following: there are many scenarios in geometry in which analogous versions of the Lefschetz hyperplane theorem hold. These scenarios are somewhat unexpected and have had a profound impact in mathematics.

math.AG↗

arXiv2019

The Lefschetz principle in birational geometry: birational twin varieties

Inspired by the Weak Lefschetz Principle, we study when a smooth projective variety fully determines the birational geometry of some of its subvarieties. In particular, we consider the natural embedding of the space of complete quadrics into the space of complete collineations and we observe that their birational geometry, from the point of view of Mori theory, fully determines each other. When two varieties are related in this way, we call them birational twins. We explore this notion and its various flavors for other embeddings between Mori dream spaces.

math.AG↗

arXiv2017

Families of elliptic curves in $\mathbb{P}^3$ and Bridgeland Stability

We study wall crossings in Bridgeland stability for the Hilbert scheme of elliptic quartic curves in three dimensional projective space. We provide a geometric description of each of the moduli spaces we encounter, including when the second component of this Hilbert scheme appears. Along the way, we prove that the principal component of this Hilbert scheme is a double blow up with smooth centers of a Grassmannian, exhibiting a completely different proof of this known result by Avritzer and Vainsencher. This description allows us to compute the cone of effective divisors of this component.

math.AG↗

arXiv2016

Extremal higher codimension cycles of the space of complete conics

Let $M$ denote the space of complete conics. We compute the cone of effective and numerically effective $k$-cycles of $M$, $\mathrm{Eff}_k(M)$ and $\mathrm{Nef}_k(M)$, respectively. In addition, we compute the Białynicki-Birula cell-decomposition of $M$ with respect to a $\mathbb{C}^*$-action and compare the cone generated by the closure of these cells to the cones $\mathrm{Eff}_k(M)$ and $\mathrm{Nef}_k(M)$.

math.AG↗

arXiv2015

Birational Geometry of the Space of Complete Quadrics

We study the birational geometry of the moduli space of complete $n$-quadrics $X$. We exhibit generators for Eff$(X)$ and Nef$(X)$, the cone of effective divisors and the cone of nef divisors, respectively. As a corollary X is a Mori Dream space. Furthermore, we run the MMP on $X$ and find a moduli interpretation for the models induced by the generators of the nef cone. In the case of complete quadric surfaces, we describe all the birational models of $X$ induced by the movable cone and find a moduli interpretation for some of these models.

math.AG↗