Asymptotic cohomological functions of toric divisors

We study functions on the class group of a toric variety measuring the rates of growth of the cohomology groups of multiples of divisors. We show that these functions are piecewise polynomial with respect to finite polyhedral chamber decompositions. As applications, we express the self-intersection number of a T-Cartier divisor as a linear combination of the volumes of the bounded regions in the corresponding hyperplane arrangement and prove an asymptotic converse to Serre vanishing.Comment: 13 pages. v2: corrected typos, minor revisions. To appear in Adv. Mat

Syzygies, multigraded regularity and toric varieties

Using multigraded Castelnuovo-Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B_1, ..., B_k on X and integers m_1, ..., m_k, consider the line bundle L := B_1^m_1 \otimes ... \otimes B_k^m_k. We give conditions on the m_i which guarantee that the ideal of X in P(H^0(X,L)) is generated by quadrics and the first p syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes.Comment: improved exposition and corrected typo

The T-graph of a multigraded Hilbert scheme

The T-graph of a multigraded Hilbert scheme records the zero and one-dimensional orbits of the T = (K^*)^n action on the Hilbert scheme induced from the T-action on A^n. It has vertices the T-fixed points, and edges the one-dimensional T-orbits. We give a combinatorial necessary condition for the existence of an edge between two vertices in this graph. For the Hilbert scheme of points in the plane, we give an explicit combinatorial description of the equations defining the scheme parameterizing all one-dimensional torus orbits whose closures contain two given monomial ideals. For this Hilbert scheme we show that the T-graph depends on the ground field, resolving a question of Altmann and Sturmfels

Cox rings and pseudoeffective cones of projectivized toric vector bundles

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 a projective space along a sequence of linear subspaces. As applications, we show that the projectivized cotangent bundles of some toric varieties are not Mori dream spaces and give examples of projectivized toric vector bundles whose Cox rings are isomorphic to that of M_{0,n}.Comment: 20 pages. v2: Revised and expanded, treating a larger class of toric vector bundles and giving new examples whose Cox rings are related to that of M_{0,n

The ring of evenly weighted points on the projective line

Let M_w = (\Pj^1)^n \q \mathrm{SL}_2 denote the geometric invariant theory quotient of (\Pj^1)^n by the diagonal action of $\mathrm{SL}_2$ using the line bundle $\mathcal{O}(w_1,w_2,...,w_n)$ on (\Pj^1)^n. Let $R_w$ be the coordinate ring of $M_w$. We give a closed formula for the Hilbert function of $R_w$, which allows us to compute the degree of $M_w$. The graded parts of $R_w$ are certain Kostka numbers, so this Hilbert function computes stretched Kostka numbers. If all the weights $w_i$ are even, we find a presentation of $R_w$ so that the ideal $I$ of this presentation has a quadratic Gr\"obner basis. In particular, $R_w$ is Koszul. We obtain this result by studying the homogeneous coordinate ring of a projective toric variety arising as a degeneration of $M_w$.Comment: 19 pages, to appear in Mathematische Zeitschrif