Projective Superspace Varieties, Superspace Quadrics and Non-Splitting

TL;DR

This paper investigates the structure of projective superspaces and their subvarieties, establishing normality for positive cases and demonstrating that certain superspace quadrics are inherently non-split, advancing understanding in superspace geometry.

Contribution

It introduces the concept of normality for positive projective superspaces and proves that smooth, non-reduced superspace quadrics are non-split, extending previous splitting problem results.

Findings

Positive projective superspaces are normal.

Smooth, non-reduced superspace quadrics are non-split.

Provides a framework for superspace subvariety analysis.

Abstract

This article is a continuation of a previous article which concerned the splitting problem for subspaces of superspaces. We begin with a general account of projective superspaces. Subsequently, we specialise to subvarieties of `positive' projective superspaces. Our main result is: positive, projective superspaces are `normal', in a sense we define. Then, among others, our main application is: smooth, non-reduced, superspace quadric hypersurfaces are non-split.