Algebraic constructions of complete m-arcs

2020/07/02 by Bartoli, Daniele, Micheli, Giacomo
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2007.00911

Abstract

Let m be a positive integer, q be a prime power, and PG(2,q) be the projective plane over the finite field \mathbb F_q. Finding complete m-arcs in PG(2,q) of size less than q is a classical problem in finite geometry. In this paper we give a complete answer to this problem when q is relatively large compared with m, explicitly constructing the smallest m-arcs in the literature so far for any m≥ 8. For any fixed m, our arcs \mathcal A_q,m satisfy |\mathcal A_q,m|-q→ -∞ as q grows. To produce such m-arcs, we develop a Galois theoretical machinery that allows the transfer of geometric information of points external to the arc, to arithmetic one, which in turn allows to prove the m-completeness of the arc.

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