The space of generically étale families

Abstract

We construct a space $G_X^n$ and a rank $n$, generically etale family of closed subspaces in a separated ambient space $X$. The constructed pair satisfies a universal property of generically etale families of closed subspaces in $X$. This universal property is derived directly from the construction and does in particular not use the Hilbert scheme. The constructed space $G_X^n$ is by its universal property canonically identified with a closed subspace of the Hilbert scheme.

28 pages; replaced $T{S}^{n-1}R\otimes R$ with $T{S}^{n-1,1}R$; simplified some proofs

References in corpus (2)

• Hilbert schemes of 8 points
• A syzygetic approach to the smoothability of zero-dimensional schemes

Cited by in corpus (1)

• Existence and properties of geometric quotients