Stability of pencils of plane curves, log canonical thresholds and multiplicities

TL;DR

This paper investigates the stability of pencils of plane curves of degree d in projective plane, linking geometric invariant theory, log canonical thresholds, and multiplicities to establish explicit stability criteria.

Contribution

It introduces new stability criteria for pencils of plane curves by relating their stability to generators, log canonical thresholds, and base point multiplicities.

Findings

Established explicit stability criteria for pencils of plane curves.

Connected stability with log canonical thresholds and base point multiplicities.

Provided a geometric invariant theory framework for classification.

Abstract

In this paper we study the problem of classifying pencils of curves of degree $d$ in $\mathbb{P}^2$ using geometric invariant theory. We consider the action of $SL(3)$ and we relate the stability of a pencil to the stability of its generators, to the log canonical threshold of its members, and to the multiplicities of its base points, thus obtaining explicit stability criteria.