Algebraic Properties of Generic Tropical Varieties

TL;DR

This paper explores how algebraic invariants like multiplicity and depth of graded ideals are reflected in the fan structure of their generic tropical varieties, revealing new geometric insights into these algebraic properties.

Contribution

It establishes a direct correspondence between multiplicity and fan structure, and shows how to infer depth and Cohen-Macaulay properties from tropical varieties.

Findings

Multiplicity corresponds to cone multiplicity in tropical varieties

Depth information can be recovered from fan structure when depth > 0

Cohen-Macaulay and almost-Cohen-Macaulay properties are detectable from tropical fans

Abstract

We show that the algebraic invariants multiplicity and depth of a graded ideal in the polynomial ring are closely connected to the fan structure of its generic tropical variety in the constant coefficient case. Generically the multiplicity of the ideal 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 the ideal from the fan structure of the generic tropical variety if the depth is known to be greater than 0. In particular, in this case we can see if the ideal is Cohen-Macaulay or almost-Cohen-Macaulay from its generic tropical variety.