3-Nets realizing a group in a projective plane
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. 140:16141624, 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. 219:672688, 2008), comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzas 3-nets (Adv. Geom. 10:287310, 2010) realizing the quaternion group of order 8 are the unique sporadic examples.
3-net projective plane dual 3-net embedding cubic curve