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Zero-dimensional subschemes of projective spaces related to double points of linear subspaces and to fattening directions


Abstract


Fix a linear subspace V\subseteq \mathbb {P}^n and a linearly independent set S\subset V. Let Z_{S,V} \subset V or Z_{s,r} with r:= \dim (V) and s=\sharp (S), be the zero-dimensional subscheme of V union of all double points 2p, p\in S, of V (not of \mathbb {P}^n if n>r). We study the Hilbert function of Z_{S,V} and of general unions in \mathbb {P}^n of these schemes. In characteristic 0 we determine the Hilbert function of general unions of Z_{2,1} (easy), of Z_{2,2} and, if n=3, general unions of schemes Z_{3,2} and Z_{2,2}

Keywords: zero-dimensional scheme; Hilbert function; postulation

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