The Hilbert Polynomial of a Symbolic Square

Abstract

Let k be an algebraically closed field, and let be a reduced closed subscheme with ideal sheaf ℐ. Let ℐ^<2> be the second symbolic power of ℐ. When C is an integral curve, we compute the Hilbert polynomial of 𝒪_ℙⁿ/ℐ^<2> in terms of invariants of C.

Key Words:

• Curve
• Hilbert polynomial
• Hypersurface
• Kähler differentials
• Singular locus
• Symbolic power

2010 Mathematics Subject Classification:

• Primary: 14N99
• Secondary: 14H50
• 14J17

ACKNOWLEDGMENTS

The problem that we address in this article came up naturally as I was working on my doctoral thesis at MIT under the direction of Bjorn Poonen. I am grateful to Prof. Poonen for his dedication and substantial help throughout the entire process. I also thank an anonymous referee for some helpful comments and suggestions. I thank Steven Kleiman for a number of references and an abundance of thorough comments. Finally, I thank Martin Kreuzer for some additional references.