k-nets embedded in a projective plane

We survey some recent results due to myself, Nicola Pace and Gábor P. Nagy. A dual k-net of order n in the finite projective plane PG(2, q) over the finite field GF (q) consists of a k ≥ 3 pairwise disjoint point-sets (components), each of size n, such that every line meeting two distinct components meets each component in precisely one point. Dual k-nets are truly combinatorial objects; nevertheless they are also of interest in Algebraic geometry and Resonance theory. Examples of dual k-nets for n > p arise naturally from affine subplanes in the dual plane of PG(2, q). On the other hand, for n < p there are known only a few examples, and we focus on this case. It has been conjectured that no dual k-net exists for k ≥ 4. This has been proven by us so far for large p compared to n, namely for p > 3 2−n) where φ is the classical Euler function. It has been conjectured that for q ≡ 1 (mod 4) there exists a unique dual 4-net, up to projectivity, namely that arising from an affine subplane of order 3, while no dual 4-net exists for for q ≡ 3 (mod 4). For k = 3 and n < p, further examples are known. An infinity family consists of dual 3-nets contained in a plane cubic curve (called algebraic dual 3-nets), another consists of dual 3-nets which are projection of pointsets lying on the edges of a tetrahedron in PG(4, q) (called tetrahedron type dual 3-nets). Let n < p. Assume that one of the loops associated with a dual 3-net is a group. We proved that then the dual 3-net is either algebraic, or of tetrahedron type, or it is sporadic of order n ∈ {8, 12, 24, 60}.