Linear General Position (i.e. Arcs) for Zero-Dimensional Schemes Over a Finite Field

Abstract

We extend some of the usual notions of projective geometry over a finite field (arcs and caps) to the case of zero-dimensional schemes defined over a finite field F_q. In particular we prove that for our type of zero-dimensional arcs the maximum degree in any r-dimensional projective space is r(q + 1) and (if either r = 2 or q is odd) all the maximal cases are projectively equivalent and come from a rational normal curve.