Arcs in Finite Projective Spaces

These notes are an outline of a course on arcs given at the Finite Geometry Summer School, University of Sussex, June 26-30, 2017. Basic objects and definitions Let K denote an arbitrary field. Let Fq denote the finite field with q elements, where q is the power of a prime p. Let Vk(K) denote the k-dimensional vector space over K. Let PGk−1(K) denote the (k − 1)-dimensional projective space over K. A projective point of PGk−1(K) is a one-dimensional subspace of Vk(K) which, with respect to a basis, is denoted by (x1, . . . , xk). The weight of a vector is the number of non-zero coordinates it has with respect to a fixed canonical basis. A k-dimensional linear code of length n and minimum distance d is a k-dimensional subspace of Vn(Fq) in which every non-zero vector has weight at least d. 1. Normal rational curve Example 1. A normal rational curve is a set of q + 1 points in PGk−1(K) projectively equivalent to S = {(1, t, . . . , tk−1) | t ∈ K ∪ {(0, . . . , 0, 1)}. Lemma 2. Any k-subset of S spans PGk−1(K). An arc S of PGk−1(K) is a subset of points with the property that any k-subset of S spans PGk−1(K). Implicitly, we will assume that S has size at least k. Date: 30 June 2017. 1