arXiv subjects

Manuel Leal

Publications and source records attributed to Manuel Leal.

4 recordsLinked to original sources

arXiv2025

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

Geometry of syzygies of sheaves on $\mathbb{P}^2$ via interpolation and Bridgeland stability

We show that the minimal free resolution of a general semi-stable sheaf $U$ on $\mathbb{P}^2$ contains a subcomplex that determines an extremal ray of the cone of effective divisors of its moduli space. We provide evidence that this is part of a general phenomenon in which minimal free resolutions, for distinct Betti tables, contain subcomplexes depending on wall-crossing. From this viewpoint, we provide new computations of the movable cones and Mori decompositions of some moduli spaces of sheaves using syzygies.

math.AG↗

arXiv2023

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↗