Publications and Preprints
Higher-dimensional Losev-Manin spaces and their geometry.
This is joint work with Patricio Gallardo, Javier Gonzalez-Anaya and Evangelos Routis.
Abstract: The classical Losev-Manin space can be interpreted as 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 proposed by Chen, Gibney, and Krashen is not a Mori dream space when the number of points is at least nine, regardless of the dimension.
Finite generation of Cox rings.
This is joint work with Antonio Laface. Published in the Notices of the American Mathematical Society.
Abstract: 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. Moreover, in this case there is a decomposition of the cone of effective divisors into polyhedral chambers that are used to study the birational geometry of the variety. 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.
Non-existence of negative curves.
This is joint work with Javier Gonzalez-Anaya and Kalle Karu. Published in International Mathematics Research Notices.
Abstract: 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.
The Fulton-MacPherson compactification is not a Mori dream space.
This is joint work with Patricio Gallardo and Evangelos Routis. Published in Mathematische Zeitschrift.
Abstract: We show that the Fulton-MacPherson compactification of the configuration space of n distinct labeled points in certain varieties of arbitrary dimension d, including projective space, is not a Mori dream space for n larger than d+8.
The geography of negative curves.
This is joint work with Javier Gonzalez-Anaya and Kalle Karu. To appear in the Michigan Mathematical Journal.
Abstract: 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.
Curves generating extremal rays in blowups of weighted projective planes.
This is joint work with Javier Gonzalez-Anaya and Kalle Karu. Published in the Journal of the London Mathematical Society.
Abstract: 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.
Generation of jets and Fujita's jet ampleness conjecture on toric varieties.
This is joint work with Zhixian Zhu. Published in the Journal of Pure and Applied Algebra.
Abstract: 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 ≥ 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.
Constructing non-Mori Dream Spaces from negative curves.
This is joint work with Javier Gonzalez-Anaya and Kalle Karu. Published in the Journal of Algebra.
Abstract: 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.
On a family of negative curves.
This is joint work with Javier Gonzalez-Anaya and Kalle Karu. Published in the Journal of Pure and Applied Algebra.
Abstract: 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.
Balanced complexes and effective divisors on M0,n.
This is joint work with Elijah Gunther and Olivia Zhang. Published in Communications in Algebra.
Abstract: Doran, Jensen and Giansiracusa showed a bijection between homogeneous elements in the Cox ring of M0,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 M0,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 M0,n for large $n$. In the case of M0,7, we classify all such irreducible elements arising from nonsingular complexes and give an example of how irreducibility can be shown in the singular case.
Examples of non-finitely generated Cox rings.
This is joint work with Kalle Karu. Published in the Canadian Mathematical Bulletin.
Abstract: 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.
Some non-finitely generated Cox rings.
This is joint work with Kalle Karu. Published in Compositio Mathematica.
Abstract: 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 M0,n of stable n-pointed genus zero curves does not have a finitely generated Cox ring if n is at least 13.
Projectivity in algebraic cobordism.
This is joint work with Kalle Karu. Published in the Canadian Journal of Mathematics.
Abstract: 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.
Bivariant algebraic cobordism.
This is joint work with Kalle Karu. Published in Algebra & Number Theory.
Abstract: 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.
Universality of K-theory.
This is joint work with Kalle Karu.
Abstract: We prove that graded K-theory is universal among oriented Borel-Moore homology theories with a multiplicative periodic formal group law. This article builds on the result of Shouxin Dai establishing the desired universality property of K-theory for schemes that admit embeddings on smooth algebraic schemes.
Descent for algebraic cobordism.
This is joint work with Kalle Karu. Published in the Journal of Algebraic Geometry.
Abstract: 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.
Cox rings and pseudoeffective cones of projectivized toric vector bundles.
This is joint work with Milena Hering, Hendrik Süß and Sam Payne. Published in Algebra & Number Theory.
Abstract: We study projectivizations of a special class of toric vector bundles that includes cotangent bundles, whose associated Klyachko filtrations are particularly simple. For these projectivized bundles, we give generators for the cone of effective divisors and a presentation of the Cox ring as a polynomial algebra over the Cox ring of a blowup of projective space at finitely many points. These constructions yield many new examples of Mori dream spaces, as well as examples where the pseudoeffective cone is not polyhedral. In particular, we show that the projectivized cotangent bundles of some toric varieties are not Mori dream spaces.
Okounkov bodies on projectivizations of rank two toric vector bundles.
Published in the Journal of Algebra.
Abstract: 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.
Projectivized rank two toric vector bundles are Mori dream spaces.
Published in Communications in Algebra.
Abstract: We prove that the Cox ring of the projectivization P(E) of a rank two toric vector bundle E, over a toric variety X, is a finitely generated k-algebra. As a consequence, P(E) is a Mori dream space if the toric variety X is projective and simplicial.