Picard groups for tropical toric schemes · arXivDesk
1709.03130Sep 10, 201714 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
From any monoid scheme X (also known as an F1-scheme) one can pass to a semiring scheme (a generalization of a tropical scheme) XS by scalar extension to an idempotent semifield S
Nearby in the stack
. 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(XS)
(and
Pic(XK)
) are isomorphic. In particular, if
XC
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(XS)
of Cartier divisors modulo principal Cartier divisors for a cancellative semiring scheme