Searcharxiv⌕ Search

arXiv subjects

Francisco Monserrat

Publications and source records attributed to Francisco Monserrat.

At least 19 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

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 $\pi: Z \rightarrow Z_0$ 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↗
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↗
arXiv2024

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 $\mathbb{F}_\delta$. 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 $\delta \geq a$, and the Cox ring of $X$ is finitely generated whenever $\delta \geq b$. 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 $\mathbb{F}_\delta$.

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↗
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↗
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↗
arXiv2020

Foliations with isolated singularities on Hirzebruch surfaces

We study foliations $\mathcal{F}$ on Hirzebruch surfaces $S_\delta$ and prove that, similarly to those on the projective plane, any $\mathcal{F}$ can be represented by a bi-homogeneous polynomial affine $1$-form. In case $\mathcal{F}$ has isolated singularities, we show that, for $ \delta=1 $, the singular scheme of $\mathcal{F}$ does determine the foliation, with some exceptions that we describe, as is the case of foliations in the projective plane. For $\delta \neq 1$, we prove that the singular scheme of $\mathcal{F}$ does not determine the foliation. However we prove that, in most cases, two foliations $\mathcal{F}$ and $\mathcal{F}'$ given by sections $s$ and $s'$ have the same singular scheme if and only if $s'=\Phi(s)$, for some global endomorphism $\Phi $ of the tangent bundle of $S_\delta$.

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↗
arXiv2019

Quantum codes from a new construction of self-orthogonal algebraic geometry codes

We present new quantum codes with good parameters which are constructed from self-orthogonal algebraic geometry codes. Our method permits a wide class of curves to be used in the formation of these codes, which greatly extends the class of a previous paper due to Munuera, Ten\'orio and Torres. These results demonstrate that there is a lot more scope for constructing self-orthogonal AG codes than was previously known.

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↗
arXiv2017

Newton-Okounkov bodies of exceptional curve valuations

We prove that the Newton-Okounkov body of the flag $E_{\bullet}:= \left\{ X=X_r \supset E_r \supset \{q\} \right\}$, defined by the surface $X$ and the exceptional divisor $E_r$ given by any divisorial valuation of the complex projective plane $\mathbb{P}^2$, with respect to the pull-back of the line-bundle $\mathcal{O}_{\mathbb{P}^2} (1)$ is either a triangle or a quadrilateral, characterizing when it is a triangle or a quadrilateral. We also describe the vertices of that figure. Finally, we introduce a large family of flags for which we determine explicitly their Newton-Okounkov bodies which turn out to be triangular.

math.AG↗
arXiv2017

Minimal plane valuations

We consider the last value $\hatμ (ν)$ of the vanishing sequence of $H^0(L)$ along a divisorial or irrational valuation $ν$ centered at $\mathcal{O}_{\mathbb{P}^2,p}$, where $L$ resp. $p$ is a line resp. a point of the projective plane $\mathbb{P}^2$ over an algebraically closed field. This value contains, for valuations, similar information as that given by Seshadri constants for points. It is always true that $\hatμ (ν) \geq \sqrt{1 / \mathrm{vol}(ν)}$ and minimal valuations are those satisfying the equality. In this paper, we prove that the Greuel-Lossen-Shustin Conjecture implies a variation of the Nagata Conjecture involving minimal valuations (that extends the one stated in the paper "Very general monomial valuations of $\mathbb{P}^2$ and a Nagata type conjecture" by Dumnicki et al. to the whole set of divisorial and irrational valuations of the projective plane) which also implies the original one. We also provide infinitely many families of very general minimal valuations with an arbitrary number of Puiseux exponents, and an asymptotic result that can be considered as an evidence in the direction of the mentioned conjecture by Dumnicki et al.

math.AG↗
arXiv2014

The cone of curves and the Cox ring of rational surfaces given by divisorial valuations

We consider surfaces $X$ defined by plane divisorial valuations $ν$ of the quotient field of the local ring $R$ at a closed point $p$ of the projective plane $\mathbb{P}^2$ over an arbitrary algebraically closed field $k$ and centered at $R$. We prove that the regularity of the cone of curves of $X$ is equivalent to the fact that $ν$ is non positive on ${\mathcal O}_{\mathbb{P}^2}(\mathbb{P}^2\setminus L)$, where $L$ is a certain line containing $p$. Under these conditions, we characterize when the characteristic cone of $X$ is closed and its Cox ring finitely generated. Equivalent conditions to the fact that $ν$ is negative on ${\mathcal O}_{\mathbb{P}^2}(\mathbb{P}^2\setminus L) \setminus k$ are also given.

math.AG↗
arXiv2011

The Poincaré Problem, algebraic integrability and dicritical divisors

We solve the Poincaré problem for plane foliations with only one dicritical divisor. Moreover, in this case, we give an algorithm that decides whether a foliation has a rational first integral and computes it in the affirmative case. We also provide an algorithm to compute a rational first integral of prefixed genus $g\neq 1$ of any type of plane foliation $\cf$. When the number of dicritical divisors dic$(\cf)$ is larger than two, this algorithm depends on suitable families of invariant curves. When dic$(\cf) = 2$, it proves that the degree of the rational first integral can be bounded only in terms of $g$, the degree of $\cf$ and the local analytic type of the dicritical singularities of $\cf$.

math.DS↗