3-Nets realizing a group in a projective plane

Abstract

In a projective plane PG(2, K) defined over an algebraically closed field K of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky (Compos. Math. 140:1614-1624, 2004, arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky (Adv. Math. 219:672-688, 2008), comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzúa's 3-nets (Adv. Geom. 10:287-310, 2010) realizing the quaternion group of order 8 are the unique sporadic examples.If p is larger than the order of the group, the above classification holds in characteristic p > 0 apart from three possible exceptions Alt 4 , Sym 4 , and Alt 5 .Motivation for the study of finite 3-nets in the complex plane comes from the study of complex line arrangements and from resonance theory; see (Falk and Yuzvinsky in

Group-labeled light dual multinets in the projective plane

“…Consider the triangular dual subnet (α 1 (Hx), α 2 (Hx), α 3 (Hx 2 )). The action of the corresponding collineation group on m i coincides with the actions of H i ; see [8,Proposition 10]. Thus, the line containing α 3 (Hx 2 ) passes through T 1 , T 2 .…”

“…where x, y, z ∈ H ⊆ K * , |H| = m and σ is an element of the semidirect product H ⋊ σ satisfying σ 2 = (xσ) 2 = 1 for all x ∈ H. In other words, G = H ⋊ σ is isomorphic to the dihedral group of order n = 2m; see [8,Section 4.4], [12, Section 6.2]. By composing the α i 's with a projection from a point P in general position to a fixed hyperplane Σ, one obtains a dual 3-net labeled by D 2m in PG(2, K).…”

“…If Q is a group, which is an important particular case in our context, a fairly complete classification of all dual 3-nets is available for p = 0 or p > n; see [8]: Apart from four groups of smaller orders (n = 8, 12, 24, 60), a group-labeled dual 3-net is either algebraic, or of tetrahedron type. An algebraic dual 3-net has its components lying on a plane cubic Γ in P G(2, K) and is labeled by an abelian group.…”

“…For a thorough discussion on the applications of finite multinets in Algebraic geometry, especially in the study of completely reducible fibers of pencils of hypersurfaces, and also in complex line arrangements and resonance theory; see [3,4,5,8,9,10,12,15,16].…”

Light dual multinets of order six in the projective plane

“…For example, if β is a projective realization of a dihedral group of order 2m, then the component β(P i ) (i = 1, 2, 3) is contained in the union of two lines, cf. [13,Proposition 22]. In other words, the projective realizations of finite dihedral groups are of tetrahedron type.…”