Algebraic properties of generic tropical varieties

Abstract

We show that the algebraic invariants multiplicity and depth of the quotient ring $S∕I$ of a polynomial ring $S$ and a graded ideal $I\subset S$ are closely connected to the fan structure of the generic tropical variety of $I$ in the constant coefficient case. Generically the multiplicity of $S∕I$ is shown to correspond directly to a natural definition of multiplicity of cones of tropical varieties. Moreover, we can recover information on the depth of $S∕I$ from the fan structure of the generic tropical variety of $I$ if the depth is known to be greater than $0$. In particular, in this case we can see if $S∕I$ is Cohen–Macaulay or almost-Cohen–Macaulay from the generic tropical variety of $I$.