2018/04/25 by Torger Olson, Olson, Torger, Eric Swartz +1
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO
paper · pdf · doi:10.48550/arxiv.1804.09707
21 pages; updated and revised
arxiv created 2018/10/03 · arxiv updated 2018/10/04
A k-arc in the projective space \rm PG(n,q) is a set of k projective points such that no subcollection of n+1 points is contained in a hyperplane. In this paper, we construct new 60-arcs and 110-arcs in \rm PG(4,q) that do not arise from rational or elliptic curves. We introduce computational methods that, when given a set P of projective points in the projective space of dimension n over an algebraic number field Q(ξ), determines a complete list of primes p for which the reduction modulo p of P to the projective space \rm PG(n,ph) may fail to be a k-arc. Using these methods, we prove that there are infinitely many primes p such that \rm PG(4,p) contains a \rm PSL(2,11)-invariant 110-arc, where \rm PSL(2,11) is given in one of its natural irreducible representations as a subgroup of \rm PGL(5,p). Similarly, we show that there exist \rm PSL(2,11)-invariant 110-arcs in \rm PG(4,p2) and \rm PSL(2,11)-invariant 60-arcs in \rm PG(4,p) for infinitely many primes p.