Some computational results on small 3-nets embedded in a projective plane over a field

TL;DR

This paper explores the embedding of specific small 3-nets in projective planes over algebraically closed fields, providing computational proofs of their algebraic nature and limitations based on the group's structure and field characteristic.

Contribution

It offers a computational approach to verify algebraic realizations of dual 3-nets for certain groups and demonstrates the non-realizability of the group Alt_4 in characteristic zero.

Findings

Dual 3-nets for C_3 x C_3 and C_2 x C_4 are algebraic, with points on a cubic.

Computational proofs confirm algebraic nature of these 3-nets.

Alt_4 group cannot be realized in characteristic zero.

Abstract

In this paper, we investigate dual 3-nets realizing the groups $C_3 \times C_3$, $C_2 \times C_4$, $\Alt_4$ and that can be embedded in a projective plane $PG(2,\mathbb K)$, where $\mathbb K$ is an algebraically closed field. We give a symbolically verifiable computational proof that every dual 3-net realizing the groups $C_3 \times C_3$ and $C_2 \times C_4$ is algebraic, namely, that its points lie on a plane cubic. Moreover, we present two computer programs whose calculations show that the group $\Alt_4$ cannot be realized if the characteristic of $\mathbb K$ is zero.