3-Nets realizing a group in a projective plane

Abstract

In a projective plane $\mathit{PG}(2,\mathbb{K})$ defined over an algebraically closed field $\mathbb{K}$ 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. Urzua'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 $\rm{Alt}_{4}$ , $\rm{Sym}_{4}$ , and $\rm{Alt}_{5}$ . 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 Compos. Math. 143:1069---1088, 2007; Miguel and Buzunariz in Graphs Comb. 25:469---488, 2009; Pereira and Yuzvinsky in Adv. Math. 219:672---688, 2008; Yuzvinsky in Compos. Math. 140:1614---1624, 2004; Yuzvinsky in Proc. Am. Math. Soc. 137:1641---1648, 2009).