Multiple Lines of Maximum Genus in $\mathbb{P}^3$

TL;DR

This paper introduces a new cohomology concept for multiple lines in projective 3-space, classifies such lines up to multiplicity 4, and reveals that certain space curves with maximal genus are not irreducible for degrees 4 and 5.

Contribution

It defines good cohomology for multiple lines in P^3 and classifies these lines up to multiplicity 4, providing new insights into the irreducibility of space curves with maximal genus.

Findings

Good cohomology notion for multiple lines introduced

Classification of multiple lines up to multiplicity 4 achieved

Space curves of degree 4 and 5 with maximal genus are not irreducible

Abstract

We introduce a notion of good cohomology for multiple lines in $\mathbb{P}^3$ and we classify multiple lines with good cohomology up to multiplicity 4. In particular, we show that the family of space curves of degree d, not lying on a surface of degree less than d, and of maximal arithmetic genus is not irreducible already for d=4 and d=5.