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The Hilbert polynomial of a symbolic square

Kaloyan Slavov

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 $C$.

The Hilbert polynomial of a symbolic square

Abstract

Let be an algebraically closed field, and let be a reduced closed subscheme with ideal sheaf . Let be the second symbolic power of . When is an integral curve, we compute the Hilbert polynomial of in terms of invariants of .

Paper Structure

This paper contains 6 sections, 9 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. An exact sequence for $\mathcal{I}^{<2>}$
  3. The Hilbert polynomial of $\mathcal{O}_{\mathbb{P}^n}/\mathcal{I}^{<2>}$
  4. The case when $C$ is a curve
  5. An explicit example
  6. Further study

Key Result

Proposition \oldthetheorem

The Hilbert polynomial of $\mathcal{O}_{\mathbb{P}^n}/\mathcal{I}^{<2>}$ is given by for $l\gg 0.$

Theorems & Definitions (21)

  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Corollary \oldthetheorem
  • Example \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Corollary \oldthetheorem
  • proof
  • proof : Proof of Proposition \ref{['Hilb_poly_O_mod_J']}.
  • ...and 11 more