The Hilbert polynomial of a symbolic square

TL;DR

This paper computes the Hilbert polynomial of the second symbolic power of the ideal sheaf of an integral curve in projective space, linking it to invariants of the curve.

Contribution

It provides an explicit formula for the Hilbert polynomial of the symbolic square of the ideal sheaf of an integral curve.

Findings

Derived the Hilbert polynomial in terms of curve invariants

Connected symbolic powers to geometric properties of curves

Extended understanding of symbolic powers in algebraic geometry

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 $C$.