arXiv subjects

Cesar Lozano Huerta

Publications and source records attributed to Cesar Lozano Huerta.

4 recordsLinked to original sources

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↗

arXiv2020

On the position of nodes of plane curves

The Severi variety $V_{d,n}$ of plane curves of a given degree $d$ and exactly $n$ nodes admits a map to the Hilbert scheme $\mathbb{P}^{2[n]}$ of zero-dimensional subschemes of $\mathbb{P}^2$ of degree $n$. This map assigns to every curve $C\in V_{d,n}$ its nodes. For some $n$, we consider the image under this map of many known divisors of the Severi variety and its partial compactification. We compute the divisor classes of such images in Pic$(\mathbb{P}^{2[n]})$ and provide enumerative numbers of nodal curves. We also answer directly a question of Diaz-Harris about whether the canonical class of the Severi variety is effective.

math.AG↗

arXiv2019

On the birational geometry of Hilbert schemes of points and Severi divisors

We study the birational geometry of Hilbert schemes of points on non-minimal surfaces. In particular, we study the weak Lefschetz Principle in the context of birational geometry. We focus on the interaction of the stable base locus decomposition (SBLD) of the cones of effective divisors of $X^{[n]}$ and $Y^{[n]}$, when there is a birational morphism $f:X\rightarrow Y$ between surfaces. In this setting, $N^1(Y^{[n]})$ embeds in $N^1(X^{[n]})$, and we ask if the restriction of the stable base locus decomposition of $N^1(X^{[n]})$ yields the respective decomposition in $N^1(Y^{[n]})$ $i.e.$, if the weak Lefschetz Principle holds. Even though the stable base loci in $N^1(X^{[n]})$ fails to provide information about how the two decompositions interact, we show that the restriction of the augmented stable base loci of $X^{[n]}$ to $Y^{[n]}$ is equal to the stable base locus decomposition of $Y^{[n]}$. We also exhibit effective divisors induced by Severi varieties. We compute the classes of such divisors and observe that in the case that $X$ is the projective plane, these divisors yield walls of the SBLD for some cases.

math.AG↗

arXiv2016

Curvas algebraicas y la pregunta de Halphen

This is an expository paper written in Spanish. This paper discusses the answer and ideas around the Halphen question: what are the pairs (g,d) that occur as the genus and degree of a smooth algebraic curve in projective 3-space. Halphen's question has been studied by many authors, including Halphen in 1881, Castelnuovo in 1890, and was finally answered almost 100 years later by Gruson and Peskin in 1981. We (very briefly) sketch arguments by Castelnuovo in studying this question and comment on the ideas of the final answer provided by Gruson and Peskin. This paper is meant to be brief and elementary introduction to some ideas about algebraic curves embedded in projective space and as such, we start out with basic definitions such as that of an algebraic curve, genus and degree. Towards the end, we mention some modern developments about this question as well as directions of research.

math.HO↗