## Publications and Preprints

Projectivity in algebraic cobordism.

This is joint work with Kalle Karu.

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.

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. (To appear in the Journal of Algebraic Geometry)

This is joint work with Kalle Karu.

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. (Appeared in Algebra & Number Theory)

This is joint work with Milena Hering, Hendrik Süß and Sam Payne.

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. (Appeared 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. (Appeared 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.