Castelnuovo-Mumford regularity bound for smooth threefolds in ℙ⁵ and extremal examples

Abstract

Let X be a nondegenerate integral subscheme of dimension n and degree d in ℙ^N defined over the complex number field ℂ. X is said to be k-regular if Hⁱ(ℙ^N, ℐ_X (k – i)) = 0 for all i ≧ 1, where ℐ_X is the sheaf of ideals of ℐ_ℙN and Castelnuovo-Mumford regularity reg(X) of X is defined as the least such k. There is a well-known conjecture concerning k-regularity: reg(X) ≦ deg(X) – codim(X) + 1. This regularity conjecture including the classification of borderline examples was verified for integral curves (Castelnuovo, Gruson, Lazarsfeld and Peskine), and an optimal bound was also obtained for smooth surfaces (Pinkham, Lazarsfeld). It will be shown here that reg(X) ≦ deg(X) – 1 for smooth threefolds X in ℙ⁵ and that the only extremal cases are the rational cubic scroll and the complete intersection of two quadrics. Furthermore, every smooth threefold X in ℙ⁵ is k-normal for all k ≧ deg(X) – 4, which is the optimal bound as the Palatini 3-fold of degree 7 shows. The same bound also holds for smooth regular surfaces in ℙ⁴ other than for the Veronese surface.