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Hilbert Schemes of Degree Four Curves

Published online by Cambridge University Press:  04 December 2007

Scott Nollet  and
Enrico Schlesinger
Scott Nollet
Affiliation:
Department of Mathematics, Texas Christian University, Fort Worth, TX 76129, USA. e-mail: s.nollet@tcu.edu
Enrico Schlesinger
Affiliation:
Dipartimento di Matematica, Politecnico di Milano, 20133 Milan, Italy. e-mail: enrsch@mate.polimi.it
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Abstract

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In this paper we determine the irreducible components of the Hilbert schemes H4,g of locally Cohen-Macaulay space curves of degree four and arbitrary arithmetic genus g: there are roughly ∼(g2/24) of them, most of which are families of multiplicity structures on lines. We give deformations which show that these Hilbert schemes are connected. For g≤−3 we exhibit a component that is disjoint from the component of extremal curves and use this to give a counterexample to a conjecture of Aït-Amrane and Perrin.

Keywords

Hilbert schemes of locally Cohen–Macaulay curvesmultiplicity structures on curves
Type
Research Article
Information
Compositio Mathematica , Volume 139 , Issue 2 , November 2003 , pp. 169 - 196
Copyright
© 2003 Kluwer Academic Publishers