Light dual multinets of order six in the projective plane (Q2338582)

Let \(\mathbb{K}\) be a field, \(Q\) a quasigroup, and for \(i = 1, 2, 3\), let \(\alpha_i : Q \rightarrow \mathrm{PG}(2,\mathbb{K})\) be maps such that the points \(\alpha_1(x), \alpha_2(y)\) and \(\alpha_3(x \cdot y)\) are collinear for all \(x, y \in Q\). Define the multisets \(\Lambda_i = \alpha_i(Q), \ i = 1, 2, 3\). Then \((\Lambda_1, \Lambda_2, \Lambda_3)\) is a dual multinet, labeled by \(Q\). If the maps \(\alpha_i\) are injective and their images \(\Lambda_i\) are disjoint, then the dual multinet is called light. If a line \(\ell\) intersects two components \(\Lambda_i\), \(\Lambda_j\) then there is an integer \(r\) such that \(r = |\ell \cap \Lambda_1| = |\ell \cap \Lambda_2| = |\ell \cap \Lambda_3|\); this integer \(r\) is called the length of \(\ell\) with respect to \((\Lambda_1, \Lambda_2, \Lambda_3)\). According to the authors, previous work suggests that ``the length \(r>1\) of lines of the light dual multinet makes a big difference in their geometric structure. While for \(r \geq 9\), the light dual multinet is well structured in (the) geometric and algebraic sense, the case of small \(r\), especially \(r=2\) shows many irregularities''. In this paper, the authors classify all abstract light dual multinets of order 6 with a unique line of length \(r>1\), and they compute all possible realizations of these abstract light dual multinets in projective planes over fields of characteristic zero.