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

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

2010-10-12

arxiv.org/abs/1010.2399v2

Mathematics

Algebraic Geometry

17 pages, revised version

Scientific paper

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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