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 of 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