Well-Poised Hypersurfaces

TL;DR

This paper classifies all well-poised hypersurfaces over algebraically closed fields, exploring their tropical varieties and Newton-Okounkov bodies, thus advancing understanding of their algebraic and geometric properties.

Contribution

It provides a complete classification of well-poised hypersurfaces and analyzes their tropical and Newton-Okounkov structures, a novel comprehensive study in this area.

Findings

Complete classification of well-poised hypersurfaces

Analysis of tropical varieties associated with these hypersurfaces

Study of Newton-Okounkov bodies related to the hypersurfaces

Abstract

An ideal $I$ is said to be "well-poised" if all of the initial ideals obtained from points in the tropical variety $Trop(I)$ are prime. This condition was first defined by Nathan Ilten and the third author. We classify all well-poised hypersurfaces over an algebraically closed field. We also study the tropical varieties and associated Newton-Okounkov bodies of these hypersurfaces.