The Hilbert schemes of locally Cohen-Macaulay curves in P^3 may after all be connected

TL;DR

This paper proves that the Hilbert schemes of locally Cohen-Macaulay curves in P^3 are connected by constructing specific families of curves, answering a longstanding open question in algebraic geometry.

Contribution

It provides a positive answer to Hartshorne's question by constructing flat families of curves with desired properties, demonstrating connectedness of the Hilbert scheme.

Findings

Constructed families of curves connecting disjoint lines to extremal curves.

Showed that every effective divisor in a smooth quadric surface lies in the same connected component.

Confirmed the connectedness of Hilbert schemes for locally Cohen-Macaulay curves in P^3.

Abstract

Progress on the problem whether the Hilbert schemes of locally Cohen-Macaulay curves in projective 3 space are connected has been hampered by the lack of an answer to a question that was raised by Robin Hartshorne in his paper "On the connectedness of the Hilbert scheme of curves in projective 3 space" Comm. Algebra 28 (2000) and more recently in the open problems list of the 2010 AIM workshop Components of Hilbert Schemes available at http://aimpl.org/hilbertschemes: does there exist a flat irreducible family of curves whose general member is a union of d disjoint lines on a smooth quadric surface and whose special member is a locally Cohen-Macaulay curve in a double plane? In this paper we give a positive answer to this question: for every d, we construct a family with the required properties, whose special fiber is an extremal curve in the sense of Martin-Deschamps and Perrin. From this we conclude that every effective divisor in a smooth quadric surface is in the connected component of its Hilbert scheme that contains extremal curves.