Classification of partitions of all triples on ten points into copies of Fano and affine planes

We continue our study of partitions of the full set of fenced(frac(v, 3)) triples chosen from a v-set into copies of the Fano plane PG (2, 2) (Fano partitions) or copies of the affine plane AG (2, 3) (affine partitions) or into copies of both of these planes (mixed partitions). The smallest cases for which such partitions can occur are v = 8 where Fano partitions exist, v = 9 where affine partitions exist, and v = 10 where both affine and mixed partitions exist. The Fano partitions for v = 8 and the affine partitions for v = 9 and 10 have been fully classified, into 11, two and 77 isomorphism classes, respectively. Here we classify (1) the sets of i pairwise disjoint affine planes for i = 1, ..., 7, and (2) the mixed partitions for v = 10 into their 22 isomorphism classes. We consider the ways in which these partitions relate to the large sets of AG (2, 3). Crown Copyrigh