Geometry of Dots and Ropes

Abstract

An a-dot is the first infinitesimal neighbourhood of a point with respect to an (a 1)-dimensional affine space. We define a notion of uniform position for a collection of dots in projective space, which in particular holds for a collection of dots arising as a general plane section of a higher-dimensional scheme. We estimate the Hilbert function of such a collection of dots, with the result that Theorem 1. Let r be a collection of d a-dots in uniform position in Pn, a > 2. Then the Hilbert function hr of F satisfies hr(r) ? min(rn + 1, 2d) + (a 2) min((r l)n 1, d) for r > 3. Equality occurs for some r with rn + 2 < 2d if and only if Fred is contained in a rational normal curve C, and the tangent directions to this curve at these points are all contained in F. Equality occurs for some r with (r 1 )n < d if and only if r is contained in thefirst infinitesimal neighbourhood of C with respect to a subbundle, of rank a 1 and of maximal degree, of the normal bundle of C in P'n. This implies an upper bound on the degree of a subbundle of rank a 1 of the normal bundle of an irreducible nondegenerate smooth curve of degree d in P'n , by a Castelnuovo argument.