Ana Maria Botero | Searcharxiv

arXiv2026

Asymptotic Behavior of Tropical Rank Functions

We show that the asymptotic behavior of the two main competing notions of rank of a linear series on a tropical curve is governed by asymptotic invariants, closely paralleling the theory of volumes in algebraic geometry. We introduce and study tropical notions of volume associated to both divisors and tropical modules. We prove optimal asymptotic results for each case. In addition, we show that the tropical volume is compatible with the tropicalization of curves.

math.AG↗

arXiv2020

The convex-set algebra and the toric b-Chow group

In \cite{botero}, a top intersection product of toric b-divisors on a smooth complete toric variety is defined. It is shown that a nef toric b-divisor corresponds to a convex set and that its top intersection number equals the volume of this convex set. The goal of this article is to extend this result and define an intersection product of sufficiently positive toric b-classes of arbitrary codimension. For this, we extend the polytope algebra of McMullen (\cite{McM1}, \cite{McM2}, \cite{McM3}) to the so called \emph{convex-set algebra} and we show that it embeds in the toric b-Chow group. In this way, the convex-set algebra can be viewed as a ring for an intersection theory for sufficiently positive toric b-classes. As an application, we show that some Hodge type inequalities are satisfied for the convex-set algebra.

math.AG↗