Free divisors in a pencil of curves

TL;DR

This paper characterizes free divisors in a pencil of plane curves, showing they are precisely unions of all singular members with a Jacobian ideal as a local complete intersection.

Contribution

It provides a complete criterion for when a union of curves in a pencil forms a free divisor, linking geometric and algebraic conditions.

Findings

Free divisors in a pencil contain all singular members.

Jacobian ideal must be a local complete intersection.

Characterization applies to curves with the same degree and smooth base locus.

Abstract

A plane curve on a the projective space over a field of characteristic zero is free if its associated sheaf T of tangent vector fields tangent is a free module. Relatively few free curves are known. Here we prove that a divisor consisting of a union of curves of a pencil of plane projective curves with the same degree and with a smooth base locus is a free divisor if and only this union contains all the singular members of the pencil and its Jacobian ideal is locally a complete intersection.