Generalized ovals in PG(3n - 1, q, with q odd

Abstract

In 1954 Segre proved that every oval of PG(2; q), with q odd, is a nonsingular conic. The proof relies on the "Lemma of Tangents". A generalized oval of PG(3n - 1; q) is a set of qn + 1 (n - 1)-dimensional subspaces of PG(3n - 1; q), every three of them generate PG(3n - 1; q); a generalized oval with n = 1 is an oval. The only known generalized ovals are essentially ovals of PG(2; qn) interpreted over GFq). If the oval of PG(2; qn) is a conic, then we call the corresponding generalized oval classical. Now assume q odd. In the paper we prove several properties of classical generalized ovals. Further we obtain a strong characterization of classical generalized ovals in PG(3n-1; q) and an interesting theorem on generalized ovals in PG(5; q), developing a theory in the spirit of Segre's approach. So for example a "Lemma of Tangents" for generalized ovals is obtained. We hope such an approach will lead to a classification of all generalized ovals in PG(3n - 1; q), with q odd.