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Joaquim Roé

Publications and source records attributed to Joaquim Roé.

At least 19 recordsLinked to original sources
arXiv2026

b-divisorial valuations and Berkovich positivity functions

We prove semicontinuity properties for local positivity invariants of big and nef divisors. The usual definition of Seshadri constant and asymptotic order of vanishing along a subvariety is extended to include all seminorms in the Berkovich space, and we obtain semicontinuity of such constants as a function of the center seminorm. We use Shokurov's language of b-divisors; to each seminorm there is an associated b-divisor which can be used to translate questions about positivity into questions about the shape of certain cones of b-divisors. The theory works especially well for what we call b-divisorial valuations, a natural extension of the notion of divisorial valuations which encompasses, e.g., all Abhyankar valuations.

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arXiv2025

Boundedness Results for Planar Linear Systems Assuming The Segre-Harbourne-Gimigliano-Hirschowitz Conjecture

Let $X_n$ be the projective plane blown up at $n \geq 10$ general points. In this paper we give several consequences of the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture, that pertain to complete linear systems on $X_n$. We begin by classifying such systems $|C|$ with general irreducible member of genus $g \geq 2$ (up to Cremona equivalence), in terms of invariants of the adjoint systems $|C+mK|$. We then use this to prove that, for fixed $n \geq 10$ and $g\geq 2$, up to the action of the Cremona group, there exist finitely many complete linear systems on $X_n$ whose general member is irreducible of genus $g$. Further, there is a function $g\mapsto n(g)$ such that every such (effective) system is Cremona equivalent to a system in $X_{n(g)}$. The latter result is based on the explicit computation of the minimum possible self-intersection of an irreducible linear system with given $n$ and $\dim(|C|)$. We classify those systems which achieve the minimal self-intersection. We also classify the systems with $C^2 \leq 5$, whether or not they have minimal $C^2$ for the given $n$ and dimension. We finish by proving several statements concerning systems that are base-point-free, and systems that give birational maps to their image.

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arXiv2024

Newton-Okounkov bodies and Picard numbers on surfaces

We study the shapes of all Newton-Okounkov bodies $Δ_{v}(D)$ of a given big divisor $D$ on a surface $S$ with respect to all rank 2 valuations $v$ of $K(S)$. We obtain upper bounds for, and in many cases we determine exactly, the possible numbers of vertices of the bodies $Δ_{v}(D)$. The upper bounds are expressed in terms of Picard numbers and they are birationally invariant, as they do not depend on the model $\tilde{S}$ where the valuation $v$ becomes a flag valuation. We also conjecture that the set of all Newton-Okounkov bodies of a single ample divisor $D$ determines the Picard number of $S$, and prove that this is the case for Picard number 1, by an explicit characterization of surfaces of Picard number 1 in terms of Newton-Okounkov bodies.

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arXiv2023

Quadric cones on the boundary of the Mori cone for very general blowups of the plane

We show the existence of cones over 8-dimensional rational spheres at the boundary of the Mori cone of the blow-up of the plane at $s\geq 13$ very general points. This gives evidence for De Fernex's strong $Δ$-conjecture, which is known to imply Nagata's conjecture. This also implies the existence of a multitude of good and wonderful rays as defined in Ciliberto-Harbourne-Miranda-Roé 2013.

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arXiv2022

Polynomial factorization over henselian fields

Given a valued field $(K,v)$ and an irreducible polynomial $g\in K[x]$, we survey the ideas of Ore, Maclane, Okutsu, Montes, Vaquié and Herrera-Olalla-Mahboub-Spivakovsky, leading (under certain conditions) to an algorithm to find the factorization of $g$ over a henselization of $(K,v)$.

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arXiv2022

Valuative trees over valued fields

For an arbitrary valued field $(K,v)$ and a given extension $v(K^*)\hookrightarrowΛ$ of ordered groups, we analyze the structure of the tree formed by all $Λ$-valued extensions of $v$ to the polynomial ring $K[x]$. As an application, we find a model for the tree of all equivalence classes of valuations on $K[x]$ (without fixing their value group), whose restriction to $K$ is equivalent to $v$. In the henselian case, we apply these results to show that there is a complete parallelism between the arithmetic properties of irreducible polynomials $F\in K[x]$, encoded by their Okutsu frames, and the valuation-theoretic properties of their induced valuations $v_F$ on $K[x]$, encoded by their MacLane-Vaquié chains. This parallelism was only known for defectless irreducible polynomials.

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arXiv2022

Valuations with infinite limit-depth

For a certain field $K$, we construct a valuation-algebraic valuation on the polynomial ring $K[x]$, whose Maclane--Vaquié chain consists of an infinite (countable) number of limit augmentations

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arXiv2022

On the number of vertices of Newton--Okounkov polygons

The Newton--Okounkov body of a big divisor D on a smooth surface is a numerical invariant in the form of a convex polygon. We study the geometric significance of the shape of Newton--Okounkov polygons of ample divisors, showing that they share several important properties of Newton polygons on toric surfaces. In concrete terms, sides of the polygon are associated to some particular irreducible curves, and their lengths are determined by the intersection numbers of these curves with D. As a consequence of our description we determine the numbers k such that D admits some k-gon as a Newton--Okounkov body, elucidating the relationship of these numbers with the Picard number of the surface, which was first hinted at by work of Küronya, Lozovanu and Maclean.

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arXiv2020

Invariants of limit key polynomials

Let $ν$ be a valuation of arbitrary rank on the polynomial ring $K[x]$ with coefficients in a field $K$. We prove comparison theorems between MacLane-Vaquié key polynomials for valuations $μ\leν$ and abstract key polynomials for $ν$. Also, some results on invariants attached to limit key polynomials are obtained. In particular, if $\operatorname{char}(K)=0$ we show that all limit key polynomials of unbounded continuous MacLane chains have numerical character equal to one.

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arXiv2019

Harbourne constants, pull-back clusters and ramified morphisms

We describe the effect of ramified morphisms on Harbourne constants of reduced effective divisors. With this goal, we introduce the pullback of a weighted cluster of infinitely near points under a dominant morphism between surfaces, and describe some of its basic properties. As an application, we describe configurations of curves with transversal intersections and $H$-index arbitarily close to $-25/7\simeq -3.571$, smaller than any previously known result.

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arXiv2019

Concave transforms of filtrations and rationality of Seshadri constants

We show that the subgraph of the concave transform of a multiplicative filtration on a section ring is the Newton--Okounkov body of a certain semigroup, and if the filtration is induced by a divisorial valuation, then the associated graded algebra is the algebra of sections of a concrete line bundle in higher dimension. We use this description to give a rationality criterion for certain Seshadri constants. Along the way we introduce Newton--Okounkov bodies of abstract graded semigroups and determine conditions for their slices to be Newton--Okounkov bodies of subsemigroups.

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arXiv2018

The 21 reducible polars of Klein's quartic

We describe the singularities and related properties of the arrangement of 21 reducible polars of Klein's quartic, containing Klein's well-known arrangement of $21$ lines.

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arXiv2017

Galois descent for the gonality of curves

We determine conditions for the invariance of the gonality under base extension, depending on the numeric invariants of the curve. More generally, we study the Galois descent of morphisms of curves to Brauer-Severi varieties, and also of rational normal scrolls.

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arXiv2017

Nagata type statements

Nagata solved Hilbert's 14-th problem in 1958 in the negative. The solution naturally lead him to a tantalizing conjecture that remains widely open after more than half a century of intense efforts. Using Nagata's theorem as starting point, and the conjecture, with its multiple variations, as motivation, we explore the important questions of finite generation for invariant rings, for support semigroups of multigraded algebras, and for Mori cones of divisors on blown up surfaces, and the rationality of Waldschimdt constants. Finally we suggest a connection between the Mori cone of the Zariski-Riemann space and the continuity of the Waldschmidt constant as a function on the space of valuations.

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arXiv2016

The Halphen cubics of order two

For each $m\ge 1$, Roulleau and Urzúa give an implicit construction of a configuration of $4(3m^2-1)$ complex plane cubic curves. This construction was crucial for their work on surfaces of general type. We make this construction explicit by proving that the Roulleau-Urzúa configuration consists precisely of the Halphen cubics of order $m$, and we determine specific equations of the cubics for $m=1$ (which were known) and for $m=2$ (which are new).

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arXiv2016

Very general monomial valuations of $\mathbb{P}^2$ and a Nagata type conjecture

It is well known that multi-point Seshadri constants for a small number $s$ of points in the projective plane are submaximal. It is predicted by the Nagata conjecture that their values are maximal for $s\geq 9$ points. Tackling the problem in the language of valuations one can make sense of $s$ points for any positive real $s\geq 1$. We show somewhat surprisingly that a Nagata-type conjecture should be valid for $s\geq 8+1/36$ points and we compute explicitly all Seshadri constants (expressed here as the asymptotic maximal vanishing element) for $s\leq 7+1/9$.

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arXiv2016

Local positivity in terms of Newton-Okounkov bodies

In recent years, the study of Newton-Okounkov bodies on normal varieties has become a central subject in the asymptotic theory of linear series, after its introduction by Lazarsfeld-Mustata and Kaveh-Khovanskii. One reason for this is that they encode all numerical equivalence information of divisor classes (by work of Jow). At the same time, they can be seen as local positivity invariants, and Küronya-Lozovanu have studied them in depth from this point of view. We determine what information is encoded by the set of all Newton-Okounkov bodies of a big divisor with respect to flags centered at a fixed point of a surface, by showing that it determines and is determined by the numerical equivalence class of the divisor up to negative components in the Zariski decomposition that do not go through the fixed point.

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