Algebraic properties of generic tropical varieties

Römer, Tim; Schmitz, Kirsten

doi:10.2140/ant.2010.4.465

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Algebraic properties of generic tropical varieties

Tim Römer and Kirsten Schmitz

Vol. 4 (2010), No. 4, 465–491

DOI: 10.2140/ant.2010.4.465

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 $I$ .

Keywords

tropical variety, constant coefficient case, Gröbner fan, generic initial ideals, Cohen–Macaulay, multiplicity, depth

Mathematical Subject Classification 2000

Primary: 13F20

Secondary: 14Q99, 13P10

Milestones

Received: 11 September 2009

Revised: 5 February 2010

Accepted: 6 April 2010

Published: 13 June 2010

Authors

Tim Römer
Institut für Mathematik Universität Osnabrück 49069 Osnabrück Germany

Kirsten Schmitz
Institut für Mathematik Universität Osnabrück 49069 Osnabrück Germany

Vol. 4, No. 4, 2010