Extending pseudo-arcs in odd characteristic

A pseudo-arc   in PG(3n−1,q)PG(3n−1,q) is a set of (n−1)(n−1)-spaces such that any three of them span the whole space. A pseudo-arc of size qn+1qn+1 is a pseudo-oval  . If a pseudo-oval OO is obtained by applying field reduction to a conic in PG(2,qn)PG(2,qn), then OO 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)PG(2,q) and we extend this construction to that of a GQ of order (qn−1,qn+1)(qn−1,qn+1) from a pseudo-arc of PG(3n−1,q)PG(3n−1,q) of size qnqn.The main theorem of this paper shows that if KK is a pseudo-arc in PG(3n−1,q)PG(3n−1,q), q   odd, of size larger than the size of the second largest complete arc in PG(2,qn)PG(2,qn), where for one element KiKi of KK, the partial spread S={K1,…,Ki−1,Ki+1,…,Ks}/KiS={K1,…,Ki−1,Ki+1,…,Ks}/Ki extends to a Desarguesian spread of PG(2n−1,q)PG(2n−1,q), then KK is contained in a pseudo-conic. The main result of Casse et al. (1985) [5] also follows from this theorem.