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 C3×C3C3×C3, C2×C4C2×C4, or Alt4Alt4. We give a symbolically verifiable computational proof that every dual 3-net realizing the groups C3×C3C3×C3 and C2×C4C2×C4 is algebraic, namely, that its points lie on a plane cubic. Moreover, we present a computational approach for showing that the group Alt4Alt4 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.