Linear system of geometrically irreducible plane cubics over finite fields

TL;DR

This paper investigates the maximum possible dimension of linear systems of plane cubic curves over finite fields where all members are geometrically irreducible, providing computational evidence and partial theoretical results towards a conjecture.

Contribution

It establishes the existence of a 3-dimensional linear system with at most one reducible member, advancing understanding of the structure of such systems over finite fields.

Findings

Maximum dimension likely 3 based on computational evidence

Existence of a 3-dimensional system with at most one reducible member proven

Supports conjecture on the maximal dimension of such systems

Abstract

We examine the maximum dimension of a linear system of plane cubic curves whose $\mathbb{F}_q$-members are all geometrically irreducible. Computational evidence suggests that such a system has a maximum (projective) dimension of $3$. As a step towards the conjecture, we prove that there exists a $3$-dimensional linear system $\mathcal{L}$ with at most one geometrically reducible $\mathbb{F}_q$-member.