On regular {v,n}-arcs in finite projective spaces

A regular {v, n}-arc of a projective space P of order q is a set S of v points such that each line of P has exactly 0,1 or n points in common with S and such that there exists a line of P intersecting S in exactly n points. Our main results are as follows: (1) If P is a projective plane of order q and if S is a regular {v, n}-arc with n ≥ √q + 1, then S is a set of n collinear points, a Baer subplane, a unital, or a maximal arc. (2) If P is a projective space of order q and if S is a regular {v, n}-arc with n ≥ √q + 1 spanning a subspace U of dimension at least 3, then S is a Baer subspace of U, an affine space of order q in U, or S equals the point set Of U. © 1993 John Wiley & Sons, Inc.