Intersections of Hypersurfaces and Ring of Conditions of a Spherical Homogeneous Space

TL;DR

This paper extends the BKK theorem to the ring of conditions of spherical homogeneous spaces and introduces the concept of rings of complete intersections, linking them to polytope volumes.

Contribution

It provides a new version of the BKK theorem for spherical homogeneous spaces and defines rings of complete intersections for these spaces and arbitrary varieties.

Findings

Proves a BKK-type theorem for spherical homogeneous spaces.

Introduces the notion of rings of complete intersections.

Describes the ring of complete intersections via volumes of polytopes.

Abstract

We prove a version of the BKK theorem for the ring of conditions of a spherical homogeneous space $G/H$. We also introduce the notion of ring of complete intersections, firstly for a spherical homogeneous space and secondly for an arbitrary variety. Similarly to the ring of conditions of the torus, the ring of complete intersections of $G/H$ admits a description in terms of volumes of polytopes.