Extending pseudo-arcs in odd characteristic

A pseudo-arc in PG(3n− 1, q) is a set of (n− 1)-spaces such that any three of them span the whole space. A pseudo-arc of size qn + 1 is a pseudo-oval. If a pseudo-oval O is obtained by applying field reduction to a conic in PG(2, qn), then O is called a pseudo-conic. We first explain the connection of (pseudo-)arcs with Laguerre planes, orthogonal arrays and generalised quadrangles. In particular, we prove that the Ahrens-Szekeres GQ is obtained from a q-arc in PG(2, q) and we extend this construction to that of a GQ of order (qn − 1, qn + 1) from a pseudo-arc of PG(3n− 1, q) of size qn. The main theorem of this paper shows that if K is a pseudo-arc in PG(3n − 1, q), q odd, of size larger than the size of the second largest complete arc in PG(2, qn), where for one element Ki of K, the partial spread S = {K1, . . . ,Ki−1,Ki+1, . . . ,Ks}/Ki extends to a Desarguesian spread of PG(2n−1, q), then K is contained in a pseudoconic. The main result of [5] also follows from this theorem.