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

Abstract

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