Powers of complete intersections: graded Betti numbers and applications

Original · EN

Let I = (F₁,...,Fᵣ) be a homogeneous ideal of R = k[x₀,...,xₙ] generated by a regular sequence of type (d₁,...,dᵣ). We give an elementary proof for an explicit description of the graded Betti numbers of Iˢ for any s ≥ 1. These numbers depend only upon the type and s. We then use this description to: (1) write Hᵣ/ᵢₛ, the Hilbert function of R/Iˢ, in terms of Hᵣ/ᵢ; (2) verify that the k-algebra R/Iˢ satisfies a conjecture of Herzog-Huneke-Srinivasan; and (3) obtain information about the numerical invariants associated to sets of fat points in Pⁿ whose support is a complete intersection or a complete intersection minus a point.