On the Embedding of (k,p) -Arcs is MaximalArcs

Abstract

Our main result is that a (k,p)-arc in \PG (2,1),q = p^h , with k \geq qp - q + p - \frac{1}{2} \sqrt[4]{q} can be extended to a maximal arc. Combining this result with the recent Ball, Blokhuis, Mazzocca theorem about the non-existence of maximal arcs for p > 2, it gives an upper bound for the size of a (k,p)-arc. The method can be regarded as a generalization of B. Segre‘s method for proving similar embeddability theorems for k-arcs (that is when n= 2). It is based on associating an algebraic envelope containing the short lines to the (k,p)-arc. However, the construction of the envelope is independent of Segre‘s method using the generalization of Menelaus‘ theorem.