The Hilbert polynomial of a symbolic square

Abstract

Let $k$ be an algebraically closed field, and let $C\subset \mathbb{P}^n_k$ be a reduced closed subscheme with ideal sheaf $\mathcal{I}$. Let $\mathcal{I}^{<2>}$ be the second symbolic power of $\mathcal{I}$. When $C$ is an integral curve, we compute the Hilbert polynomial of $\mathcal{O}_{\mathbb{P}^n}/\mathcal{I}^{<2>}$ in terms of invariants of $C$.