Sam Payne

Nonarchimedean geometry, tropicalization, and metrics on curves

We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and a generalization of the Sturmfels-Tevelev multiplicity formula in tropical elimination theory to the case of a nontrivial valuation. For curves, we explore in detail the relationship between skeletal metrics and lattice lengths on tropicalizations and show that the maps from the analytification of a curve to the tropicalizations of its toric embeddings stabilize to an isometry on finite subgraphs. Other applications include generalizations of Speyers well-spacedness condition and the Katz-Markwig-Markwig results on tropical j-invariants.

Toric vector bundles, branched covers of fans, and the resolution property

We associate to each toric vector bundle on a toric variety X(∆) a “branched cover” of the fan ∆ together with a piecewise-linear function on the branched cover. This construction generalizes the usual correspondence between toric Cartier divisors and piecewise-linear functions. We apply this combinatorial geometric technique to investigate the existence of resolutions of coherent sheaves by vector bundles, using singular nonquasiprojective toric threefolds as a testing ground. Our main new result is the construction of complete toric threefolds that have no nontrivial toric vector bundles of rank less than or equal to three. The combinatorial geometric sections of the paper, which develop a theory of cone complexes and their branched covers, can be read independently.

Cayley decompositions of lattice polytopes and upper bounds for h*-polynomials

Abstract We give an effective upper bound on the h*-polynomial of a lattice polytope in terms of its degree and leading coefficient, confirming a conjecture of Batyrev. We deduce this bound as a consequence of a strong Cayley decomposition theorem which says, roughly speaking, that any lattice polytope with a large multiple that has no interior lattice points has a nontrivial decomposition as a Cayley sum of polytopes of smaller dimension. Polytopes with nontrivial Cayley decompositions correspond to projectivized sums of toric line bundles, and our approach is partially inspired by classification results of Fujita and others in algebraic geometry. In an appendix, we interpret our Cayley decomposition theorem in terms of adjunction theory for toric varieties.

Ehrhart Series and Lattice Triangulations

Abstract We express the generating function for lattice points in a rational polyhedral cone with a simplicial subdivision in terms of multivariate analogues of the h-polynomials of the subdivision and “local contributions” of the links of its nonunimodular faces. We also compute new examples of nonunimodal h*-vectors of reflexive polytopes.

Tropical independence I: Shapes of divisors and a proof of the Gieseker–Petri theorem

We develop a framework to apply tropical and nonarchimedean analytic techniques to multiplication maps on linear series and study degenerations of these multiplications maps when the special fiber is not of compact type. As an application, we give a tropical criterion for a curve over a valued field to be Gieseker-Petri general.

Moduli of toric vector bundles

We give a presentation of the moduli stack of toric vector bundles with fixed equivariant total Chern class as a quotient of a fine moduli scheme of framed bundles by a linear group action. This fine moduli scheme is described explicitly as a locally closed subscheme of a product of partial flag varieties cut out by combinatorially specified rank conditions. We use this description to show that the moduli of rank three toric vector bundles satisfy Murphy’s law, in the sense of Vakil. The preliminary sections of the paper give a selfcontained introduction to Klyachko’s classification of toric vector bundles.

Tropical independence, II: The maximal rank conjecture for quadrics

Building on our earlier results on tropical independence and shapes of divisors in tropical linear series, we give a tropical proof of the maximal rank conjecture for quadrics. We also prove a tropical analogue of Max Noethers theorem on quadrics containing a canonically embedded curve, and state a combinatorial conjecture about tropical independence on chains of loops that implies the maximal rank conjecture for algebraic curves.

Realization Spaces for Tropical Fans

We introduce a moduli functor for varieties whose tropicalization realizes a given weighted fan and show that this functor is an algebraic space in general, and is represented by a scheme when the associated toric variety is quasiprojective. We study the geometry of these tropical realization spaces for the matroid fans studied by Ardila and Klivans, and show that the tropical realization space of a matroid fan is a torus torsor over the realization space of the matroid. As a consequence, we deduce that these tropical realization spaces satisfy Murphy’s Law.

Lifting divisors on a generic chain of loops

Let C be a curve over a complete valued field having an infinite residue field and whose skeleton is a chain of loops with generic edge lengths. We prove that any divisor on the chain of loops that is rational over the value group lifts to a divisor of the same rank on C , confirming a conjecture of Cools, Draisma, Robeva, and the third author.

Topology of nonarchimedean analytic spaces and relations to complex algebraic geometry

This note surveys basic topological properties of nonarchimedean analytic spaces, in the sense of Berkovich, including the recent tameness results of Hrushovski and Loeser. We also discuss interactions between the topology of nonarchimedean analytic spaces and classical algebraic geometry.