An upper bound on the number of rational points of arbitrary projective varieties over finite fields

We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field. This bound depends only on the dimensions and degrees of the irreducible components and holds for very general varieties, even reducible and non equidimensional. As a consequence, we prove a conjecture of Ghorpade and Lachaud on the maximal number of rational points of an equidimensional projective variety

Maximal partial spreads and transversal-free translation nets

AbstractIn this paper, we are concerned with three topics in finite geometry which have generated much interest in the literature: maximal partial spreads (or t-spreads), translation nets, and maximal sets of mutually orthogonal Latin squares. We obtain large maximal sets of MOLS for infinitely many new parameter pairs by constructing appropriate transversal-free translation nets which in turn belong to suitable maximal partial spreads. To this purpose, we first obtain some improved bounds on translation nets and partial t-spreads and then give a detailed study of the possible extensions of a translation net of small or critical deficiency. In order to apply these results, we also construct maximal partial t-spreads with previously unknown parameters. Given any prime power q = pa, where p is a prime ⩾5, we show the existence of a transversal-free translation net of order q2 (and hence a maximal set of MOLS of order q2) for each of the deficiencies d = q – 1, q, and q + 1. Finally, using results of Evans, we also obtain interesting examples of rather small transversal-free translation nets; in particular, we determine all transversal-free translation nets of order p2 and degree p + 1 for a prime p with abelian translation group