3-Nets realizing a group in a projective plane |
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Authors: | Gábor Korchmáros Gábor P. Nagy Nicola Pace |
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Affiliation: | 1. Dipartimento di Matematica e Informatica, Università della Basilicata, Contrada Macchia Romana, 85100, Potenza, Italy 2. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720, Szeged, Hungary 4. MTA-ELTE Geometric and Algebraic Combinatorics Research Group, Pázmány Péter sétány 1/C, 1117, Budapest, Hungary 3. Inst. de Ciências Matemáticas e de Computa??o, Universidade de S?o Paulo, Av. do Trabalhador S?o-Carlense, 400, S?o Carlos, SP, 13560-970, Brazil
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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. 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 $rm{Alt}_{4}$ , $rm{Sym}_{4}$ , 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 Buzunáriz 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). |
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