Reduced superschemes and the combinatorics of toric supervarieties

TL;DR

This paper introduces new foundational definitions for supergeometry, extends classical results to supervarieties, and explores the combinatorial structure of toric supervarieties using decorated fans and polytopes.

Contribution

It establishes a rigorous framework for supervarieties, generalizes classical algebraic geometry results, and links toric supervarieties to combinatorial objects like decorated fans and polytopes.

Findings

Equivalence between certain toric supervarieties and decorated polyhedral fans

Decorated fans encode key geometric information about toric supervarieties

Toric supervarieties inside the isomeric supergrassmannian can be described as decorated polytopes

Abstract

We propose new definitions of integral, reduced, and normal superrings and superschemes to properly establish the notion of a supervariety. We generalize several results about classical reduced rings and varieties to the supergeometric setting, including an equivalence of categories between certain toric supervarieties and decorated polyhedral fans. These decorated fans are shown to encode important geometric information about the corresponding toric supervarieties. We then investigate some naturally-occurring toric supervarieties inside the isomeric supergrassmannian, which we show admits a nice description as a decorated polytope.