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

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.