Powers of complete intersections: graded Betti numbers and applications

Abstract

Let $I = (F_1,\ldots,F_r)$ be a homogeneous ideal of the ring $R = k[x_0,\ldots,x_n]$ generated by a regular sequence of type $(d_1,\ldots,d_r)$. We give an elementary proof for an explicit description of the graded Betti numbers of $I^s$ for any $s \geq 1$. These numbers depend only upon the type and $s$. We then use this description to: (1) write $H_{R/I^s}$, the Hilbert function of $R/I^s$, in terms of $H_{R/I}$; (2) verify that the $k$-algebra $R/I^s$ satisfies a conjecture of Herzog-Huneke-Srinivasan; and (3) obtain information about the numerical invariants associated to sets of fat points in $\mathbb{P}^n$ whose support is a complete intersection or a complete intersection minus a point.