The Hilbert scheme of space curves sitting on a smooth surface containing a line

TL;DR

This paper investigates the structure and properties of families of smooth space curves on smooth surfaces containing a line, extending known results and identifying conditions for irreducibility, smoothness, and non-reducedness.

Contribution

It extends the classification of maximal families of space curves on surfaces with a line, including new results on irreducibility, smoothness, and non-reducedness for various degrees and genera.

Findings

Extended the ranges where families are irreducible and generically smooth.

Identified non-reduced components for specific surface degrees s=4 and 5.

Progressed towards understanding non-reducedness in families for s > 3.

Abstract

We continue the study of maximal families W of the Hilbert scheme, H(d,g)_{sc}, of smooth connected space curves whose general curve C lies on a smooth degree-s surface S containing a line. For s > 3, we extend the two ranges where W is a unique irreducible (resp. generically smooth) component of H(d,g)_{sc}. In another range, close to the boarder of the nef cone, we describe for s=4 and 5 components W that are non-reduced, leaving open the non-reducedness of only 3 (resp. 2) families for s > 5 (resp. s=5), thus making progress to recent results of Kleppe and Ottem in [28]. For s=3 we slightly extend previous results on a conjecture of non-reduced components, and in addition we show its existence in a subrange of the conjectured range.