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

TL;DR

This paper characterizes non-singular quadrics and Hermitian varieties in finite classical polar spaces through their intersection numbers with hyperplanes and codimension 2 spaces, extending previous results and providing criteria for non-singularity.

Contribution

It generalizes prior characterizations by Ferri and Tallini, offering necessary and sufficient conditions for identifying non-singular quadrics and Hermitian varieties via intersection properties.

Findings

Complete characterization of non-singular quadrics and Hermitian varieties.

Extension of Ferri and Tallini's results to broader classes.

Criteria for non-singularity based on intersection numbers.

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 \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.