Completeness of cubic curves in PG(2, q), q <= 81

TL;DR

This paper systematically determines the completeness of all cubic curves in the projective plane PG(2,q) for q up to 81, extending known theoretical results to specific finite fields.

Contribution

It provides a complete classification of the completeness of cubic curves in PG(2,q) for q ≤ 81, filling a gap in the existing theoretical understanding.

Findings

All cubic curves in PG(2,q) for q ≤ 81 are classified as complete or incomplete.

The results confirm and extend theoretical predictions for small q.

The classification aids in understanding the structure of (n,3)-arcs in finite projective planes.

Abstract

Theoretical results are known about the completeness of a planar algebraic cubic curve as a (n,3)-arc in PG(2,q). They hold for q big enough and sometimes have restriction on the characteristic and on the value of the j-invariant. We determine the completeness of all cubic curves for q <= 81.