Pencils of plane cubics with one base point

TL;DR

This paper classifies pencils of plane cubics with a single base point and smooth general members, establishing uniqueness under irreducibility and comparing with classical methods.

Contribution

It provides a complete classification of such pencils and proves the uniqueness of the non-isotrivial case with irreducible members.

Findings

Complete classification of pencils with one base point

Uniqueness of non-isotrivial pencils with irreducible members

Comparison with classical algebraic geometry approaches

Abstract

We study pencils of plane cubics with only one base point and general member smooth, giving a complete classification. Under the additional hypothesis that all members are irreducible, we prove that there exists a unique non-isotrivial pencil with these properties up to projective transformation. We compare our construction with the classical approaches given by Gattazzo, Beauville and Miranda-Persson.