3-Nets realizing a group in a projective plane

Author(s)

Korchmaros, Gabor ^[1] ; Nagy, Gabor P. ^{[2, 3]} ; Pace, Nicola ^[4]

Affiliation

^[1] Univ Basilicata, Dipartimento Matemat & Informat, I-85100 Potenza - Italy

^[2] Univ Szeged, Bolyai Inst, H-6720 Szeged - Hungary

^[3] MTA ELTE Geometr & Algebra Combinator Res Grp, H-1117 Budapest - Hungary

^[4] Univ Sao Paulo, Inst Ciencias Matemat & Comp, BR-13560970 Sao Carlos, SP - Brazil

• Document type: Journal article
• Source: JOURNAL OF ALGEBRAIC COMBINATORICS; v. 39, n. 4, p. 939-966, JUN 2014.
• Web of Science Citations: 5

Abstract

In a projective plane defined over an algebraically closed field 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. UrzA(0)a'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 , , and . 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). (AU)

• FAPESP's process: 12/03526-0 - Finite geometry, Algebraic curves and Applications to Coding Theory
• Grantee: Nicola Pace
• Support Opportunities: Scholarships in Brazil - Post-Doctoral