On small 3-nets embedded in a projective plane over a field

In this paper, we investigate the projective embeddings of dual 3-nets realizing the groups C 3 × C 3 , C 2 × C 4 , or Alt 4 . We give a symbolically verifiable computational proof that every dual 3-net realizing the groups C 3 × C 3 and C 2 × C 4 is algebraic, namely, that its points lie on a plane cubic. Moreover, we present a computational approach for showing that the group Alt 4 cannot be realized if the characteristic of the ground field is zero. These results are fundamental for the complete classification of 3-nets embedded in a projective plane over a field.