k-nets embedded in a projective plane over a field

Korchmáros, G. and Nagy, Gábor Péter and Pace, N. (2015) k-nets embedded in a projective plane over a field. COMBINATORICA, 35 (1). pp. 63-74. ISSN 0209-9683

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Abstract

We investigate k-nets with k ≥ 4 embedded in the projective plane P G(2, K) defined over a field K; they are line configurations in P G(2, K) consisting of k pairwise disjoint line-sets, called compo- nents, such that any two lines from distinct families are concurrent with exactly one line from each component. The size of each com- ponent of a k-net is the same, the order of the k-net. If K has zero characteristic, no embedded k-net for k ≥ 5 exists; see [10, 13]. Here we prove that this holds true in positive characteristic p as long as p is sufficiently large compared with the order of the k-net. Our ap- proach, different from that used in [10, 13], also provides a new proof in characteristic zero.

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