Hilbert Functions, Residual Intersections, and Residually S2 Ideals

Let R be a homogeneous ring over an infinite field, IR a homogeneous ideal, and I an ideal generated by s forms of degrees d₁,...,d_s so that codim(:I)s. We give broad conditions for when the Hilbert function of R/ or of R/(:I) is determined by I and the degrees d₁,...,d_s. These conditions are expressed in terms of residual intersections of I, culminating in the notion of residually S₂ ideals. We prove that the residually S₂ property is implied by the vanishing of certain Ext modules and deduce that generic projections tend to produce ideals with this property.