The Hilbert polynomial of a symbolic square

Abstract

Let $k$ be an algebraically closed field, and let $C\subset {\mathbb{P}}_k^n$ 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 $C$.

Previous version has been restructured and the title has been changed. A number of suggestions for revision have been incorporated. The essential mathematical core content remains the same as in the earlier version. Section 5 with an explicit example has been added. A reference for Prop. 2.1 has been added

References in corpus (3)

• Linear subspaces, symbolic powers and Nagata type conjectures
• Star configurations in ${\mathbb{P}}^n$
• The moduli space of hypersurfaces whose singular locus has high dimension