Picard groups for tropical toric schemes

Abstract

From any monoid scheme X (also known as an 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 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 ).

Vector bundles on tropical schemes

“…Remark 7.1. We remark that the above results are different from the one that we obtained in [JMT19], where we prove that for a monoid scheme X satisfying Condition 4.10, one has the following isomorphisms:…”

“…In this section, we briefly recall basic definitions and properties of monoid schemes and semiring schemes (also tropical schemes as a special case) which play a key role in the later sections. Most of the material in this section can be found in [GG16], [Jun17a], [JMT19], [JRT20].…”

“…In [JMT19], it is shown that if X is an irreducible monoid scheme satisfying Condition 4.10, then one has the following natural isomorphisms: 12 Pic(X ) ≃ Pic(X K ) ≃ Pic(X k ), where K is an idempotent semifield, and k is a field. One may ask whether or not a similar result is true for vector bundles, but obviously one cannot have the above correspondence with vector bundles.…”

“…In [JMT19], we explored several properties of Picard groups for tropical schemes. Our main motivation was to take the first step toward formulating a scheme-theoretic tropical Riemann-Roch theorem.…”