${\mathbb{A}}^{0|1}$-torsors, quotients by free ${\mathbb{A}}^{0|1}$-actions, and embeddings into $\Pi$-projective spaces and super-Grassmannians $G(1|1,n|n)$

TL;DR

This paper investigates conditions for embedding superschemes into specific supergeometric spaces, utilizing torsors and fibrations, and establishes the existence of quotients under certain group actions.

Contribution

It introduces criteria for embeddability into $ ext{Pi}$-projective spaces and supergrassmannians, based on ${ m A}^{0|1}$-torsors and fibrations, and proves the existence of quotients for free ${ m A}^{0|1}$ actions.

Findings

Criteria for embeddability into supergeometric spaces.

Existence of quotients for free ${ m A}^{0|1}$ actions.

Connections between torsors, fibrations, and embeddings.

Abstract

We study embeddability of superschemes into $\Pi$-projective spaces and into supergrassmannians $G(1|1,n|n)$. We give some criteria based on the relation with ${\mathbb{A}}^{0|1}$-torsors and ${\mathbb{A}}^{0|1}$-fibrations. We also prove the existence of nice quotients for free actions of ${\mathbb{A}}^{0|1}$ on superschemes.