有限经典极空间与余维为2的超平面和类的相交数刻画
Characterizations of finite classical polar spaces by intersection numbers with hyperplanes and speces of codimension 2
摘要：
本文证明了非奇异二次曲面和非奇异埃尔米特簇是由它们关于余维为2的超平面和空间的相交数完全刻画的。这强有力地推广了Ferri和Tallini \cite{FT}的一个结果，并且也提供了拟二次曲面（分别是它们的埃尔米特类似曲面）是非奇异二次曲面（分别是埃尔米特簇）的充要条件。\section{Introduction}当塞格雷\cite{Segre}证明了他著名的特点二次曲线（“每一套$q+1$点在$\mathrm{PG}(2,q)$，$q$奇，其中没有三个是共线，是一个圆锥”），他没有更多的证明一个美丽和有趣的定理，他实际上提供了起点的一个新的方向组合几何。在这个分支的组合的想法是提供纯粹的组合特征的对象经典定义的代数方式。本文试图通过证明经典有限极空间的强特征来对这一理论做出贡献。
英文摘要：
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.
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