Phase limit sets of linear spaces and discriminants

TL;DR

This paper characterizes the phase limit sets of linear spaces and discriminants using coamoebas, Bergman fans, and polyhedral descriptions, providing new insights into their geometric structure and relationships.

Contribution

It introduces a novel description of phase limit sets of discriminants via coamoebas and flags of flats, linking them to Bergman fans and polyhedral geometry.

Findings

Closure of coamoeba is a union of products from flags of flats.

Partial description of phase limit sets of discriminants and duals of toric varieties.

3D components of phase limit sets are prisms over 2D discriminant coamoebas.

Abstract

We show that the closure of the coamoeba of a linear space/hyperplane complement is the union of products of coamoebas of hyperplane complements coming from flags of flats, and relate this to the Bergman fan. Using the Horn-Kapranov parameterization of a reduced discriminant, this gives a partial description of the phase limit sets of discriminants and duals of toric varieties. When d=3, we show that each 3-dimensional component of the phase limit set of the discriminant is a prism over a discriminant coamoeba in dimension 2, which has a polyhedral description by a result of Nilsson and Passare.