Basecondary polytopes

TL;DR

This paper introduces basecondary polytopes, a new class encompassing various important polytopes in geometry, including secondary, base, and Newton polytopes, providing a unified framework for their study.

Contribution

The paper defines and explores the class of basecondary polytopes, unifying several important polytopes in algebraic and combinatorial geometry, including those related to discriminants and ramified coverings.

Findings

Introduces the concept of basecondary polytopes.

Shows how they unify various classes of polytopes.

Includes the discriminant of the Lyashko--Looijenga map.

Abstract

Many (if not most) of convex polytopes, important for combinatorial and algebraic geometry, are closely related to secondary polytopes of point configurations, or base polytopes of submodular functions, or their numerous variations and generalizations. The aim of this text is to introduce the class of basecondary polytopes. This class includes (and allows to study uniformly) the aforementioned ones, as well as some others, e.g. appearing as Newton polytopes of important discriminant hypersurfaces. Most notably, this includes the discriminant of the Lyashko--Looijenga map, which is important for enumerative geometry of ramified coverings and cannot be reduced (by far) to Gelfand--Kapranov--Zelevinsky's A-discriminants and secondary polytopes.