Publications and Preprints

On a family of negative curves.                                    
            This is joint work with Javier Gonzalez-Anaya and Kalle Karu          (19 pages, 9 figures).

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 $\overline{M}_{0,n}$.                                    
            This is joint work with Elijah Gunther and Olivia Zhang          (17 pages, 8 figures).

Abstract:  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.

Examples of non-finitely generated Cox rings.                                    
            This is joint work with Kalle Karu          (20 pages, 3 figures).

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.

            Posters:             Poster Operational Cobordism.              Poster Cox Rings PTVB.              Poster Okounkov Bodies PTVB.

            Thesis:               PhD Thesis at UM: Toric Projective Bundles.