On the smooth locus of aligned Hilbert schemes: the k-secant lemma and the general projection theorem

TL;DR

This paper investigates the smoothness of aligned Hilbert schemes of finite subschemes on smooth varieties, proving that the locus of non-smooth points is contained in a proper subset of the ambient projective space.

Contribution

It establishes the structure of the smooth locus of aligned Hilbert schemes and introduces the k-secant lemma and general projection theorem.

Findings

The non-smooth locus does not fill the entire projective space.

Lines carrying the non-smooth locus are contained in a proper subset.

The expected dimension of the Hilbert scheme aligns with theoretical predictions.

Abstract

Let X be a smooth, connected, dimension n, quasi-projective variety imbedded in \PP_N. Consider integers {k_1,...,k_r}, with k_i>0, and the Hilbert Scheme H_{k_1,...,k_r}(X) of aligned, finite, degree \sum k_i, subschemes of X, with multiplicities k_i at points x_i (possibly coinciding). The expected dimension of H_{k_1,...,k_r}(X) is 2N-2+r-(\sum k_i)(N-n). We study the locus of points where H_{k_1,...,k_r}(X) is not smooth of expected dimension and we prove that the lines carrying this locus do not fill up \PP_N