arXiv subjects

Elvira Pérez-Callejo

Publications and source records attributed to Elvira Pérez-Callejo.

5 recordsLinked to original sources

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

Linear weighted bounded negativity

We propose a linear version of the weighted bounded negativity conjecture. It considers a smooth projective surface $X$ over an algebraically closed field of characteristic zero and predicts the existence of a common lower bound on $C^2/(D\cdot C)$ for all reduced and irreducible curves $C$ and all big and nef divisors such that $D\cdot C>0$, both on $X$. We prove that, in the complex case, there exists such a bound for all nef divisors spanning a ray out an open covering of the limit rays of negative curves. In the same vein, we provide explicit bounds when $X$ is a rational surface. Our proofs involve the existence of a foliation $\mathcal{F}$ on $X$ but most of our results are independent of $\mathcal{F}$.

math.AG↗

arXiv2024

Algebraic integrability with bounded genus

We provide an algorithm which decides whether a polynomial foliation $\mathcal{F}^{\mathbb{C}^2}$ on the complex plane has a polynomial first integral of genus $g\neq 1$. Except in a specific case, an extension of the algorithm also decides if $\mathcal{F}^{\mathbb{C}^2}$ has a rational first integral of that genus.

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↗

arXiv2022

Algebraic integrability of planar polynomial vector fields by extension to Hirzebruch surfaces

We study algebraic integrability of complex planar polynomial vector fields $X=A (x,y)(\partial/\partial x) + B(x,y) (\partial/\partial y) $ through extensions to Hirzebruch surfaces. Using these extensions, each vector field $X$ determines two infinite families of planar vector fields that depend on a natural parameter which, when $X$ has a rational first integral, satisfy strong properties about the dicriticity of the points at the line $x=0$ and of the origin. As a consequence, we obtain new necessary conditions for algebraic integrability of planar vector fields and, if $X$ has a rational first integral, we provide a region in $\mathbb{R}_{\geq 0}^2$ that contains all the pairs $(i,j)$ corresponding to monomials $x^i y^j$ involved in the generic invariant curve of $X$.

math.AG↗