Higher Infinitesimal Neighbourhoods

We study the Hilbert function of thekth order neighbourhood of a collection Γ of points in linearly general position in n, writtenh n(Γk, m). Our main results are the following bounds onh n(Γk, k + 1): If deg Γ ≥ 2n − k + 1 thenh n(Γk, k + 1) ≥ h n(Ck, k + 1), whereC nis a rational normal curve of degreen. If Γ is sufficiently general ands = deg Γ then and equality holds whenk = 3.