Picard groups for tropical toric schemes

Abstract

From any monoid scheme $X$ (also known as an ${\mathbb{F}}_1$-scheme) one can pass to a semiring scheme (a generalization of a tropical scheme) $X_S$ by scalar extension to an idempotent semifield $S$. We prove that for a given irreducible monoid scheme $X$ (satisfying some mild conditions) and an idempotent semifield $S$, the Picard group $Pic(X)$ of $X$ is stable under scalar extension to $S$ (and in fact to any field $K$). In other words, we show that the groups $Pic(X)$ and $Pic(X_S)$ (and $Pic(X_K)$) are isomorphic. In particular, if $X_{\mathbb{C}}$ is a toric variety, then $Pic(X)$ is the same as the Picard group of the associated tropical scheme. The Picard groups can be computed by considering the correct sheaf cohomology groups. We also define the group $CaCl(X_S)$ of Cartier divisors modulo principal Cartier divisors for a cancellative semiring scheme $X_S$ and prove that $CaCl(X_S)$ is isomorphic to $Pic(X_S)$.

14 pages, v2; (i) the title has been changed from "Picard groups for tropical toric varieties'' to "Picard groups for tropical toric schemes'', (ii) minor changes have been made following the suggestions of the referees

References in corpus (4)

• Prime congruences of idempotent semirings and a Nullstellensatz for tropical polynomials
• Picard Groups and Class Groups of Monoid Schemes
• Combinatorial tropical surfaces
• Cech Cohomology of Semiring Schemes