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José Luis González

Publications and source records attributed to José Luis González.

At least 19 recordsLinked to original sources
arXiv2026

Higher-dimensional Losev-Manin spaces and their geometry

The classical Losev-Manin space is a toric compactification of the moduli space of $n$ points in the affine line modulo translation and scaling. Motivated by this, we study its higher-dimensional toric counterparts, which compactify the moduli space of $n$ distinct labeled points in affine space modulo translation and scaling. We show that these moduli spaces are a fibration over a product of projective spaces -- with fibers isomorphic to the Losev-Manin space -- and that they are isomorphic to the normalization of a Chow quotient. Moreover, we present a criterion to decide whether the blow-up of a toric variety along the closure of a subtorus is a Mori dream space. As an application, we demonstrate that a related generalization of the moduli space of pointed rational curves constructed by Chen, Gibney, and Krashen is not a Mori dream space when the number of points is at least nine, regardless of the dimension.

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arXiv2024

Polymatroids and moduli of points in flags

We introduce and study different compactifications of the moduli space of $n$ distinct weighted labeled points in a flag of affine spaces. We construct these spaces via the weighted and generalized Fulton-MacPherson compactifications of Routis and Kim-Sato. For certain weights, our compactifications are toric and isomorphic to the polypermutohedral and polystellahedral varieties, which arise in the work of Crowley-Huh-Larson-Simpson-Wang and Eur-Larson on polymatroids, a generalization of matroids. Moreover, we show that these toric compactifications have a fibration structure, with fibers isomorphic to the Losev-Manin space, and are related to each other via a geometric quotient.

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arXiv2022

Finite generation of Cox rings

In this expository note we discuss a class of graded algebras named Cox rings, which are naturally associated to algebraic varieties generalizing the homogeneous coordinate rings of projective spaces. Whenever the Cox ring is finitely generated, the variety admits a quotient presentation by a quasitorus, which resembles the quotient construction of the projective space. We discuss the problem of the finite generation of Cox rings from a geometric perspective and provide examples of both the finitely and non-finitely generated cases.

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arXiv2021

Non-existence of negative curves

Let $X$ be a projective toric surface of Picard number one blown up at a general point. We bring an infinite family of examples of such $X$ whose Kleiman-Mori cone of curves is not closed: there is no negative curve generating one of the two boundary rays of the cone. These examples are related to Nagata's conjecture and rationality of Seshadri constants.

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arXiv2021

Generation of jets and Fujita's jet ampleness conjecture on toric varieties

Jet ampleness of line bundles generalizes very ampleness by requiring the existence of enough global sections to separate not just points and tangent vectors, but also their higher order analogues called jets. We give sharp bounds guaranteeing that a line bundle on a projective toric variety is $k$-jet ample in terms of its intersection numbers with the invariant curves, in terms of the lattice lengths of the edges of its polytope, in terms of the higher concavity of its piecewise linear function and in terms of its Seshadri constant. For example, the tensor power $k+n-2$ of an ample line bundle on a projective toric variety of dimension $n \geq 2$ always generates all $k$-jets, but might not generate all $(k+1)$-jets. As an application, we prove the $k$-jet generalizations of Fujita's conjectures on toric varieties with arbitrary singularities.

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arXiv2021

The geography of negative curves

We study the Mori Dream Space (MDS) property for blowups of weighted projective planes at a general point and, more generally, blowups of toric surfaces defined by a rational plane triangle. The birational geometry of these varieties is largely governed by the existence of a negative curve in them, different from the exceptional curve of the blowup. We consider a parameter space of all rational triangles, and within this space we study how the negative curves and the MDS property vary. One goal of the article is to catalogue all known negative curves and show their location in the parameter space. In addition to the previously known examples we construct two new families of negative curves. One of them is, to our knowledge, the first infinite family of special negative curves. The second goal of the article is to show that the knowledge of negative curves in the parameter space often determines the MDS property. We show that in many cases this is the only underlying mechanism responsible for the MDS property.

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arXiv2020

Curves generating extremal rays in blowups of weighted projective planes

We consider blowups at a general point of weighted projective planes and, more generally, of toric surfaces with Picard number one. We give a unifying construction of negative curves on these blowups such that all previously known families appear as boundary cases of this. The classification consists of two classes of said curves, each depending on two parameters. Every curve in these two classes is algebraically related to other curves in both classes; this allows us to find their defining equations inductively. For each curve in our classification, we consider a family of blowups in which the curve defines an extremal class in the effective cone. We give a complete classification of these blowups into Mori Dream Spaces and non-Mori Dream Spaces. Our approach greatly simplifies previous proofs, avoiding positive characteristic methods and higher cohomology.

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arXiv2018

Constructing non-Mori Dream Spaces from negative curves

We study blowups of weighted projective planes at a general point, and more generally blowups of toric surfaces of Picard number one. Based on the positive characteristic methods of Kurano and Nishida, we give a general method for constructing examples of Mori Dream Spaces and non-Mori Dream Spaces among such blowups. Compared to previous constructions, this method uses the geometric properties of the varieties and applies to a number of cases. We use it to fully classify the examples coming from two families of negative curves.

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arXiv2017

On a family of negative curves

Let $X$ be the blowup of a weighted projective plane at a general point. We study the problem of finite generation of the Cox ring of $X$. Generalizing examples of Srinivasan and Kurano-Nishida, we consider examples of $X$ that contain a negative curve of the class $H-mE$, where $H$ is the class of a divisor pulled back from the weighted projective plane and $E$ is the class of the exceptional curve. For any $m>0$ we construct examples where the Cox ring is finitely generated and examples where it is not.

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arXiv2017

Balanced complexes and effective divisors on $\overline{M}_{0,n}$

Doran, Jensen and Giansiracusa showed a bijection between homogeneous elements in the Cox ring of $\overline{M}_{0,n}$ not divisible by any exceptional divisor section, and weighted pure-dimensional simplicial complexes satisfying a zero-tension condition. Motivated by the study of the monoid of effective divisors, the pseudoeffective cone and the Cox ring of $\overline{M}_{0,n}$, we point out a simplification of the zero-tension condition and study the space of balanced complexes. We give examples of irreducible elements in the monoid of effective divisors of $\overline{M}_{0,n}$ for large $n$. In the case of $\overline{M}_{0,7}$, we classify all such irreducible elements arising from nonsingular complexes and give an example of how irreducibility can be shown in the singular case.

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arXiv2017

Examples of non-finitely generated Cox rings

We bring examples of toric varieties blown up at a point in the torus that do not have finitely generated Cox rings. These examples are generalizations of previous work where toric surfaces of Picard number 1 were studied. In this article we consider toric varieties of higher Picard number and higher dimension. In particular, we bring examples of weighted projective 3-spaces blown up at a point that do not have finitely generated Cox rings.

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arXiv2014

Some non-finitely generated Cox rings

We give a large family of weighted projective planes, blown up at a smooth point, that do not have finitely generated Cox rings. We then use the method of Castravet and Tevelev to prove that the moduli space of stable n-pointed genus zero curves does not have a finitely generated Cox ring if n is at least 13.

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arXiv2013

Projectivity in Algebraic Cobordism

The algebraic cobordism group of a scheme is generated by cycles that are proper morphisms from smooth quasiprojective varieties. We prove that over a field of characteristic zero the quasiprojectivity assumption can be omitted to get the same theory.

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arXiv2013

Bivariant Algebraic Cobordism

We associate a bivariant theory to any suitable oriented Borel-Moore homology theory on the category of algebraic schemes or the category of algebraic G-schemes. Applying this to the theory of algebraic cobordism yields operational cobordism rings and operational G-equivariant cobordism rings associated to all schemes in these categories. In the case of toric varieties, the operational T-equivariant cobordism ring may be described as the ring of piecewise graded power series on the fan with coefficients in the Lazard ring.

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arXiv2013

Universality of K-Theory

We prove that graded K-theory is universal among oriented Borel-Moore homology theories with a multiplicative periodic formal group law.

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arXiv2013

Descent for algebraic cobordism

We prove the exactness of a descent sequence relating the algebraic cobordism groups of a scheme and its envelopes. Analogous sequences for Chow groups and K-theory were previously proved by Gillet.

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arXiv2011

Okounkov bodies on projectivizations of rank two toric vector bundles

The global Okounkov body of a projective variety is a closed convex cone that encodes asymptotic information about every big line bundle on the variety. In the case of a rank two toric vector bundle E on a smooth projective toric variety, we use its Klyachko filtrations to give an explicit description of the global Okounkov body of P(E). In particular, we show that this is a rational polyhedral cone and that P(E) is a Mori dream space.

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