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 M_0,n of stable n-pointed genus zero curves does not have a finitely generated Cox ring if n is at least 13.

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.

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.

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.

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.

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.

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.

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.