Transitive PSL(2,11)-invariant k-arcs in PG(4,q)

Abstract

A k -arc in the projective space 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 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 PG(n, p h ) may fail to be a k-arc. Using these methods, we prove that there are infinitely many primes p such that PG(4, p) contains a PSL(2, 11)-invariant 110-arc, where PSL(2, 11) is given in one of its natural irreducible representations as a subgroup of PGL (5, p). Similarly, we show that there exist PSL(2, 11)-invariant 110-arcs in PG(4, p 2 ) and PSL(2, 11)-invariant 60-arcs in PG(4, p) for infinitely many primes p.