Families of 0-dimensional submanifolds on supercurves

TL;DR

This paper proves the existence and smoothness of the Hilbert scheme of 0-dimensional subspaces on supercurves of dimension (1|1), and demonstrates that it is generally not split.

Contribution

It establishes the existence, smoothness, and non-split nature of the Hilbert scheme for 0-dimensional subspaces on supercurves of dimension (1|1).

Findings

Hilbert scheme exists for supercurves of dimension (1|1)

The Hilbert scheme is smooth

The Hilbert scheme is generally not split

Abstract

In the present paper we prove that the Hilbert scheme of 0-dimensional subspaces on supercurves of dimension $(1|1)$ exists and it is smooth. We show that the Hilbert scheme is not split in general.