  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 \cite{FT} and also provides necessary and
sufficient conditions for quasi-quadrics (respectively their Hermitian
analogues) to be non-singular quadrics (respectively Hermitian varieties).}
\section{Introduction} When Segre \cite{Segre} proved his celebrated
characterization of conics ("every set of $q+1$ points in $\mathrm{PG}(2,q)$,
$q$ 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.
