Conic linear series and pencils of plane quartics

TL;DR

This paper investigates conic linear systems on smooth complex curves in projective space, analyzing their limits and infinitesimal properties, and constructs a specific non-isotrivial pencil of plane quartics with unique geometric features.

Contribution

It introduces a systematic study of conic linear systems on curves, including their limits and geometric structures, and proves the existence of a special pencil of quartics with particular properties.

Findings

Existence of a non-isotrivial pencil of quartics with one base point

All members of the pencil are irreducible

General member of the pencil is smooth

Abstract

We study linear systems cut out by cones of fixed degree on a smooth complex curve $C\subset\mathbb{P}^{3}$. We develop a systematic study of the families of such systems, considering their limits, their infinitesimal behaviour and some associated geometric structures. As an application, we prove the existence of a non-isotrivial pencil of quartics with only one base point, all whose members are irreducible and whose general member is smooth.