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2001•arXiv (Cornell University)Open access

Infinitesimal Extensions of P^1 and their Hilbert Schemes

Nikolaos Tziolas

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Abstract

In order to calculate the multiplicity of an isolated rational curve C in a local complete intersection variety X, i.e. the length of the Hilbert scheme of X at [C], it is important to study infinitesimal neighborhoods of the curve in X. This is equivalent to infinitesimal extensions of P^1 by locally free sheaves. In this paper we study infinitesimal extensions of P^1, determine their structure and obtain upper and lower bounds for the length of the local rings of their Hilbert schemes at [P^].

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In order to calculate the multiplicity of an isolated rational curve C in a local complete intersection variety X, i.e. the length of the Hilbert scheme of X at [C], it is important to study infinitesimal neighborhoods of the curve in X. This is equivalent to infinitesimal extensions of P^1 by locally free sheaves. In this paper we study infinitesimal extensions of P^1, determine their structure and obtain upper and lower bounds for the length of the local rings of their Hilbert schemes at [P^].

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Available abstract

In order to calculate the multiplicity of an isolated rational curve C in a local complete intersection variety X, i.e. the length of the Hilbert scheme of X at [C], it is important to study infinitesimal neighborhoods of the curve in X. This is equivalent to infinitesimal extensions of P^1 by locally free sheaves. In this paper we study infinitesimal extensions of P^1, determine their structure and obtain upper and lower bounds for the length of the local rings of their Hilbert schemes at [P^].

Key concepts: Infinitesimal, Mathematics, Hilbert scheme, Multiplicity (mathematics), Pure mathematics, Intersection (aeronautics), Scheme (mathematics), Order (exchange)

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