Abstract

We study arcs K in PG(n, q), n ≥ 3, q odd, having many points common with a given normal rational curve L. In particular, we show that, if 0.09q + 2.09 ≥ n ≥ 3, q large, then (q + l)/2 is the largest possible number of points of K on L, improving on the bound given in [11], [12], [14]. When |K ∩ L| = (q + l)/2, we show that the points of K ∩ L are invariant under a cyclic linear collineation of order (q ± l)/2. The corresponding questions for q even are discussed in [13].

Introduction

Let Σ = PG(n, q) denote the n-dimensional projective space over the field GF(q). A k-arc in Σ, with k ≥ n + 1, is a set K of k points such that no n + 1 points of K belong to a hyperplane of Σ. A point r of PG(n, q) extends a k-arc K, in PG(n, q), to a (k + l)-arc if and only if K ∪ {r} is a (k + l)-arc. A k-arc K of PG(n, q) is complete if and only if K is not contained in a (k + l)-arc of PG(n, q). Otherwise, K is called incomplete.