Compositio Mathematica

Hilbert Schemes of Degree Four Curves

Scott Nollet a1 and Enrico Schlesinger a2
a1 Department of Mathematics, Texas Christian University, Fort Worth, TX 76129, USA. e-mail:
a2 Dipartimento di Matematica, Politecnico di Milano, 20133 Milan, Italy. e-mail:

Article author query
nollet n   [Google Scholar] 
schlesinger e   [Google Scholar] 


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 [similar](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[less-than-or-equal]−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.

Key Words: Hilbert schemes of locally Cohen–Macaulay curves; multiplicity structures on curves.