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

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Abstract

In this paper, we investigate the projective embeddings of dual 3-nets realizing the groups C3×C3, C2×C4, or Alt4. We give a symbolically verifiable computational proof that every dual 3-net realizing the groups C3×C3 and C2×C4 is algebraic, namely, that its points lie on a plane cubic. Moreover, we present a computational approach for showing that the group Alt4 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.

Keywords

3-Net
Dual 3-net
Embedding
Cubic curve
Groebner basis

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1

This author was a Bolyai János Research Fellow. Also supported by the European Union and co-funded by the European Social Fund; project number: TÁMOP-4.2.2.A-11/1/KONV-2012-0073.