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arXiv · 1010.2399

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

Laurent Gruson·Christian Peskine
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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

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Laurent Gruson, Christian Peskine. 2011-12-09. On the smooth locus of aligned Hilbert schemes: the k-secant lemma and the general projection theorem. https://doi.org/10.1215/00127094-2019817

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