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Rick Miranda

Publications and source records attributed to Rick Miranda.

At least 19 recordsLinked to original sources
arXiv2026

Contact Invariants for Plane Curves in a Pencil

Let $\calP$ be a general pencil of curves of degree $d$ in the projective plane. In this paper we review the computation of the number of curves in $\calP$ that have a hyperflex line, a flex bitangent line or a tritangent line. Then we focus on the curves in the dual plane described by the flex tangents and the bitangents of the curves of $\calP$ and the curves in the original plane described by the flexes and the points of bitangencies of the curves in $\calP$. Some of these curves have been studied already: we mainly focus here on the ones that still have not been treated systematically, and we compute their degree, genus, and singularities.

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arXiv2026

Bounds on the Mordell-Weil rank of elliptic fibrations

We prove that the Mordell-Weil group of a higher dimensional elliptic fibration naturally embeds into that of a suitable elliptic surface. We give sufficient conditions for the existence of such surfaces. We apply our result to obtain explicit and uniform bounds for the Mordell-Weil rank of elliptic threefolds of Kodaira dimension zero, including Calabi-Yau threefolds, confirming predictions from physics. We prove new explicit bounds for a broad class of elliptic fourfolds. These results suggest a general linear bound for the Mordell-Weil rank in terms of the dimension of the elliptic fibration, which we formulate as a conjecture.

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arXiv2026

Constructions for rational multiple planes

A finite, normal cover $f: X\longrightarrow \bbP^2$ of degree $m\geq 3$ (the case $m=2$ is well known and we do not consider it in this paper) is called \emph{simple}, if there is a pencil $\mathcal P$ of rational curves of $\bbP^2$ such that the pull back via $f$ of $\mathcal P$ is a pencil of rational curves on $X$. Up to Cremona equivalence $\mathcal P$ can be assumed to be the pencil of lines through a fixed point $p\in \bbP^2$. If $\frakB$ is the branch curve of such a multiple plane, the general line through $p$ has to intersect $\frakB$ in $2m-2$ branch points (counted with multiplicities). If $p$ is not one of these branch points, then the multiple plane is said to be \ \emph{simpler}. \ In that case the branch curve will have a point of multiplicity $\deg(\frakB)-2m+2$ at $p$. In this paper we classify, under suitable generality conditions for the branch curve, { simpler } triple planes up to Cremona equivalence (they belong to infinitely many non--Cremona equivalent families) and we give examples of infinitely many non--Cremona equivalent families of {simpler } multiple planes of degree $m\geq 4$.

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arXiv2026

Rational elliptic surfaces with six singular double fibres

A rational elliptic surface with section is a smooth, rational, complex, projective surface $\mathcal{X}$ that admits a relatively minimal fibration $f: \mathcal{X}\longrightarrow \bbP^1$ such that its general fibre is a smooth irreducible curve of genus one and $f$ has a section. In this paper, we classify rational elliptic surfaces with section that have exactly six singular fibres, each counted with multiplicity two. The fibres that appear with multiplicity exactly two are either of type $II$ or of type $I_2$ of the Kodaira classification. We interpret our classification from various viewpoints: a pencil of plane cubic curves, the Weierstrass equation, a double cover of $\bbF_2$ branched over an appropriate trisection of the ruling of $\bbF_2$ plus the negative section, a double cover of the plane branched along a quartic curve, plus the datum of a point on the plane. Moreover, either we give explicit normal forms for the plane quartic curve, or we indicate how to find it.

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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

Variations on Pascal's Theorem

In this paper we present a variety of statements that are in the spirit of the famous theorem of Pascal, often referred to as the Mystic Hexagon. We give explicit equations describing the conditions for $d+4$ points to lie on rational normal curves. A collection of problems of Pascal type are considered for quadric surfaces in $\bbP^3$. Finally we reprove, using computer algebra methods, a remarkable theorem of Richmond, Segre, and Brown, for quadrics in $\bbP^4$ containing five general lines.

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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 $\Delta$-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-Ro\'e 2013.

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arXiv2022

Coxeter theory for curves on blowups of $\mathbb{P}^r$

We investigate the study of smooth irreducible rational curves in $Y_s^r$, a general blowup of $\mathbb{P}^r$ at $s$ general points, whose normal bundle splits as a direct sum of line bundles all of degree $i$, for $i \in \{-1,0,1\}$: we call these $(i)$-curves. We systematically exploit the theory of Coxeter groups applied to the Chow space of curves in $Y_s^r$, which provides us with a useful bilinear form that helps to expose properties of $(i)$-curves. We are particularly interested in the orbits of lines (through $1-i$ points) under the Weyl group of standard Cremona transformations (all of which are $(i)$-curves): we call these $(i)$-Weyl lines. We prove various theorems related to understanding when an $(i)$-curve is an $(i)$-Weyl line, via numerical criteria expressed in terms of the bilinear form. We obtain stronger results for $r=3$, where we prove a Noether-type inequality that gives a sharp criterion.

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arXiv2021

The moduli space of rational elliptic surfaces of index two

In this paper we construct a moduli space for marked rational elliptic surfaces of index two as a non-complete toric variety of dimension nine. We also construct compactifications of this moduli space, which are obtained as quotients of $\mathbb{A}^{12}$ by an action of $\mathbb{G}_m^3$.

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arXiv2021

On $(i)$-Curves in Blowups of $\mathbb{P}^r$

In this paper we study $(i)$-curves with $i\in \{-1, 0, 1\}$ in the blown up projective space $\mathbb{P}^r$ in general points. The notion of $(-1)$-curves was analyzed in the early days of mirror symmetry by Kontsevich with the motivation of counting curves on a Calabi-Yau threefold. In dimension two, Nagata studied planar $(-1)$-curves in order to construct counterexample to Hilbert's 14th problem. We introduce the notion of classes of $(0)$- and $(1)$-curves in $\mathbb{P}^r$ with $s$ points blown up and we prove that their number is finite if and only if the space is a Mori Dream Space. We further introduce a bilinear form on a space of curves, and a unique symmetric Weyl-invariant class, $F$, (that we will refer to as the anticanonical curve class). For Mori Dream Spaces we prove that $(-1)$-curves can be defined arithmetically by the linear and quadratic invariants determined by the bilinear form. Moreover, $(0)$- and $(1)$-Weyl lines give the extremal rays for the cone of movable curves in $\mathbb{P}^r$ with $r+3$ points blown up. As an application, we use the technique of movable curves to reprove that if $F^2\leq 0$ then $Y$ is not a Mori Dream Space and we propose to apply this technique to other spaces.

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arXiv2021

Cremona Orbits in $\mathbb{P}^4$ and Applications

This article is motivated by the authors interest in the geometry of the Mori dream space $\mathbb{P}^4$ blown up in $8$ general points. In this article we develop the necessary technique for determining Weyl orbits of linear cycles for the four-dimensional case, by explicit computations in the Chow ring of the resolution of the standard Cremona transformation. In particular, we close this paper with applications to the question of the dimension of the space global sections of effective divisors having at most $8$ base points.

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arXiv2015

A few questions about curves on surfaces

In this note we address the following kind of question: let X be a smooth, irreducible, projective surface and D a divisor on X$satisfying some sort of positivity hypothesis, then is there some multiple of D depending only on X which is effective or movable? We describe some examples, discuss some conjectures and prove some results that suggest that the answer should in general be negative, unless one puts some really strong hypotheses either on D or on X.

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arXiv2012

Riemann-Roch theory on finite sets

In [1] M. Baker and S. Norine developed a theory of divisors and linear systems on graphs, and proved a Riemann-Roch Theorem for these objects (conceived as integer-valued functions on the vertices). In [2] and [3] the authors generalized these concepts to real-valued functions, and proved a corresponding Riemann-Roch Theorem in that setting, showing that it implied the Baker-Norine result. In this article we prove a Riemann-Roch Theorem in a more general combinatorial setting that is not necessarily driven by the existence of a graph.

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arXiv2011

Linear Systems on Edge-Weighted Graphs

Let R be any subring of the reals. We present a generalization of linear systems on graphs where divisors are R-valued functions on the set of vertices and graph edges are permitted to have nonegative weights in R. Using this generalization, we provide an independent proof of a Riemann-Roch formula, which implies the Riemann-Roch formula of Baker and Norine.

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