Components of the Hilbert scheme of space curves on low-degree smooth surfaces

TL;DR

This paper investigates the structure and components of the Hilbert scheme of smooth space curves on low-degree smooth surfaces, providing conditions for smoothness, describing dimensions, and identifying non-reduced components, especially on cubic surfaces.

Contribution

It offers new criteria for the smoothness and dimension of Hilbert scheme components and explicitly describes non-reduced components on surfaces with Picard number at most 2, advancing understanding of space curve families.

Findings

Conditions for smoothness of Hilbert scheme components

Explicit descriptions of non-reduced components on certain surfaces

Progress on conjecture for curves on cubic surfaces

Abstract

We study maximal families W of the Hilbert scheme, H(d,g)_{sc}, of smooth connected space curves whose general curve C lies on a smooth surface S of degree s. We give conditions on C under which W is a generically smooth component of H(d,g)_{sc} and we determine dim W. If s=4 and W is an irreducible component of H(d,g)_{sc}, then the Picard number of S is at most 2 and we explicitly describe, also for s > 4, non-reduced and generically smooth components in the case Pic(S) is generated by the classes of a line and a smooth plane curve of degree s-1. For curves on smooth cubic surfaces the first author finds new classes of non-reduced components of H(d,g)_{sc}, thus making progress in proving a conjecture for such families.