Characterizations of finite classical polar spaces by intersection numbers with hyperplanes and speces of codimension 2
Abstract
In this article we show that non-singular quadrics and non-singular Hermitian varieties are completely characterized by their intersection numbers with respect to hyperplanes and spaces of codimension 2. This strongly generalizes a result by Ferri and Tallini FT and also provides necessary and sufficient conditions for quasi-quadrics (respectively their Hermitian analogues) to be non-singular quadrics (respectively Hermitian varieties). Introduction When Segre Segre proved his celebrated characterization of conics ("every set of q+1 points in PG(2,q), q odd, no three of which are collinear, is a conic"), he did more than proving a beautiful and interesting theorem; he in fact provided the starting point of a new direction in combinatorial geometry. In this branch of combinatorics the idea is to provide purely combinatorial characterizations of objects classically defined in an algebraic way. This article wants to contribute to this theory by proving strong characterizations of classical finite polar spaces.