arXiv subjects

Carlos-Jesús Moreno-Ávila

Publications and source records attributed to Carlos-Jesús Moreno-Ávila.

9 recordsLinked to original sources

arXiv2026

On the smallest numerical semigroups closed under affine maps

We study numerical semigroups $S_{a,b}(m)$ generated by the orbit of $m$ under the affine map $T_{a,b}(z)=az+b$, where $a\ge 2$, $m>1$, $\gcd(b,m)=1$, and $b\ge -(a-2)m-2$. This extends the usual affine-closed setting to feasible negative values of $b$. We write $A_i=(a^i-1)/(a-1)$ and let $n$ be the smallest positive integer such that $A_n\ge m$. We determine the minimal generators and give an explicit description of the Apéry set, obtaining homogeneity and formulas for the Frobenius number and the genus. We also study pseudo-Frobenius numbers via the induced Apéry parametrization and prove the sharp upper bound $\operatorname{t}(S_{a,b}(m))\le n-1$ for the type. We give a complete characterization of the symmetric members of the family in terms of the canonical representative of $m-1$. Finally, we exhibit a subfamily whose pseudo-Frobenius numbers form an arithmetic progression of length $n-1$; in particular, this subfamily attains the bound.

math.AC↗

arXiv2026

Nef divisors of surfaces given by pencils at infinity

We give generators for the nef cone and the cone of curves of rational surfaces obtained by blowing-up the complex projective plane at a set of points $\mathcal{B} \cup \mathcal{D}$, where $\mathcal{B}$ is the set of (proper and infinitely near) base points of a pencil associated with a curve having one place at infinity, and $\mathcal{D}$ is a set of finitely many infinitely near free points on the strict transforms of curves of the pencil. We also prove that, when the pencil is given by an AMS-type curve and $\mathcal{D}$ contains at most two free points on any curve considered, the Cox ring of the obtained surface is finitely generated.

math.AG↗

arXiv2025

The cone of curves and the Cox ring of rational surfaces over Hirzebruch surfaces

Let $X$ be a rational surface obtained by blowing up at a configuration $\mathcal{C}$ of infinitely near points over a Hirzebruch surface $\mathbb{F}_δ$. We prove that there exist two positive integers $a \leq b$ such that the cone of curves of $X$ is finite polyhedral and minimally generated when $δ\geq a$, and the Cox ring of $X$ is finitely generated whenever $δ\geq b$. The integers $a$ and $b$ depend only on a combinatorial object (a graph decorated with arrows) representing the strict transforms of the exceptional divisors, their intersections and those with the fibers and special section of $\mathbb{F}_δ$.

math.AG↗

arXiv2024

On weighted bounded negativity for rational surfaces

The weighted bounded negativity conjecture considers a smooth projective surface $X$ and looks for a common lower bound on the quotients $C^2/(D\cdot C)^2$, where $C$ runs over the integral curves on $X$ and $D$ over the big and nef divisors on $X$ such that $D \cdot C >0$. We focus our study on rational surfaces $Z$. Setting $π: Z \rightarrow Z_0$ a composition of blowups giving rise to $Z$, where $Z_0$ is the projective plane or a Hirzebruch surface, we give a common lower bound on $C^2/(H^* \cdot C)^2$ whenever $H^*$ is the pull-back of a nef divisor $H$ on $Z_0$. In addition, we prove that, only in the case when a nef divisor $D$ on $Z$ approaches the boundary of the nef cone, the quotients $C^2/(D\cdot C)^2$ could tend to minus infinity.

math.AG↗

arXiv2022

On the valuative Nagata conjecture

We provide several equivalent conditions for a plane divisorial valuation of a smooth projective surface to be minimal with respect to an ample divisor. These conditions involve a valuative Seshadri constant and other global tools of the surface defined by the divisorial valuation. As a consequence, we derive several equivalent statements for the valuative Nagata conjecture and some related results.

math.AG↗

arXiv2022

On the degree of curves with prescribed multiplicities and bounded negativity

We provide a lower bound on the degree of curves of the projective plane $\mathbb{P}^2$ passing through the centers of a divisorial valuation $ν$ of $\mathbb{P}^2$ with prescribed multiplicities, and an upper bound for the Seshadri-type constant of $ν$, $\hatμ(ν)$, constant that is crucial in the Nagata-type valuative conjecture. We also give some results related to the bounded negativity conjecture concerning those rational surfaces having the projective plane as a relatively minimal model.

math.AG↗

arXiv2021

Seshadri-type constants and Newton-Okounkov bodies for non-positive at infinity valuations of Hirzebruch surfaces

We consider flags $E_\bullet=\{X\supset E\supset \{q\}\}$, where $E$ is an exceptional divisor defining a non-positive at infinity divisorial valuation $ν_E$ of a Hirzebruch surface $\mathbb{F}_δ$ and $X$ the surface given by $ν_E,$ and determine an analogue of the Seshadri constant for pairs $(ν_E,D)$, $D$ being a big divisor on $\mathbb{F}_δ$. The main result is an explicit computation of the vertices of the Newton-Okounkov bodies of pairs $(E_\bullet,D)$ as above, showing that they are quadrilaterals or triangles and distinguishing one case from another.

math.AG↗

arXiv2019

Discrete equivalence of non-positive at infinity plane valuations

Non-positive at infinity valuations are a class of real plane valuations which have a nice geometrical behavior. They are divided in three types. We study the dual graphs of non-positive at infinity valuations and give an algorithm for obtaining them. Moreover we compare these graphs attending the type of their corresponding valuation.

math.AG↗

arXiv2018

Non-positive and negative at infinity divisorial valuations of Hirzebruch surfaces

We consider rational surfaces $Z$ defined by divisorial valuations $ν$ of Hirzebruch surfaces. We introduce the concepts of non-positivity and negativity at infinity for these valuations and prove that these concepts admit nice local and global equivalent conditions. In particular we prove that, when $ν$ is non-positive at infinity, the extremal rays of the cone of curves of $Z$ can be explicitly given.

math.AG↗