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In this article we show that non-singular quadrics and non-singular Hermitianvarieties are completely characterized by their intersection numbers withrespect to hyperplanes and spaces of codimension 2. This strongly generalizes aresult by Ferri and Tallini \cite{FT} and also provides necessary andsufficient conditions for quasi-quadrics (respectively their Hermitiananalogues) to be non-singular quadrics (respectively Hermitian varieties).}\section{Introduction} When Segre \cite{Segre} proved his celebratedcharacterization 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 thanproving a beautiful and interesting theorem; he in fact provided the startingpoint of a new direction in combinatorial geometry. In this branch ofcombinatorics the idea is to provide purely combinatorial characterizations ofobjects classically defined in an algebraic way. This article wants tocontribute to this theory by proving strong characterizations of classicalfinite polar spaces.

Characterizations of finite classical polar spaces by intersection numbers with hyperplanes and speces of codimension 2