High-order vanishing ideals at n+3 points of Pn

Abstract

The authors conjecture that the ideal IZ[t−1] of functions vanishing to order t−1 at the subscheme Z of Pn, n=2t−1 comprised of 2t+2 generic smooth points, satisfies dimk(IZ[t−1])t=2t, in its initial degree, t. They show that this dimension is at least t+1, by a direct construction of suitable vanishing forms. This result is complementary to those of M.V. Catalisano, P. Ellia, and A. Gimigliano in [5]. The authors also consider related problems, including the Macaulay dual problem, of determining the Hilbert function H(A), A=R/(x12,…,xr2,(x1+⋯+xr)2,L2) — where R=k[x1,…,xr], r=2t and L is a generic linear form — in the socle degree t of A.