Varieties of apolar subschemes of toric surfaces

TL;DR

This paper explores the concept of apolar schemes on toric surfaces, connecting powersum varieties with apolarity via Cox rings, and provides explicit examples for two specific toric surfaces.

Contribution

It explicitly relates apolarity to Cox rings and characterizes varieties of apolar schemes on certain toric surfaces, expanding understanding of powersum varieties.

Findings

Apolar schemes can be described using Cox rings.

Explicit examples of apolar varieties on two toric surfaces are provided.

Connections between powersum varieties and apolarity are clarified.

Abstract

Powersum varieties, also called varieties of sums of powers, have provided examples of interesting relations between varieties since their first appearance in the 19th century. One of the most useful tools to study them is apolarity, a notion originally related to the action of differential operators on the polynomial ring. In this work we make explicit how one can see apolarity in terms of the Cox ring of a variety. In this way powersum varieties are a special case of varieties of apolar schemes; we explicitely describe examples of such varieties in the case of two toric surfaces, when the Cox ring is particularly well-behaved.