Castelnuovo Regularity for Smooth Subvarieties of Dimensions 3 and 4
نویسنده
چکیده
is surjective, i.e., hypersurfaces of degree k cut out a complete linear system on X . We say that X is k–regular if H(P, IX(k − i)) = 0 for all i ≥ 1, where IX is the sheaf of ideals of X in OPr . It is easy to see that X is (k + 1)– regular if and only if X is k–normal and H(X,OX(k − i)) = 0 for all i > 0. Let reg(X) = min{k ∈ Z : X is k–regular}. The importance of k–regularity stems from the following well-known results ([Mu1], lecture 14): if X is k–regular, then the saturated ideal IX is generated by homogeneous polynomials of degree at most k and hence there is no (k+1)-secant line to X . Furthermore, the Hilbert polynomial and the Hilbert function of X have the same values for all integers m ≥ k − 1. There is a well-known conjecture concerning the k–normality and k–regularity of X :
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تاریخ انتشار 1998