椭圆二次曲面的相对$m$ -卵形体
Relative $m$-ovoids of elliptic quadrics
摘要：
设${\cal Q}^-(2n+1,q)$是${\rm PG}(2n+1,q)$的椭圆二次曲面。${\cal Q}^-(2n+1,q)$的相对$m$ -卵形体（相对于抛物线部分${\cal Q} := {\cal Q}(2n,q) \subset {\cal Q}^-(2n+1,q)$）是${\cal Q}^-(2n+1,q)\setminus {\cal Q}$的点的子集$\cal R$，使得不包含在$\cal Q$中的${\cal Q}^-(2n+1,q)$的每个生成元在精确的$m$点中满足$\cal R$。相对$m$卵形体与其互补体（在${\cal Q}^-(2n+1,q) \setminus {\cal Q}$中）具有相同的大小，称为相对半球。我们表明，一个非平凡的相对$m$ -卵形的${\cal Q}^-(2n+1,q)$必然是一个相对半球，迫使$q$甚至。构造了${\cal Q}^-(4n+1,q)$的相对半系统$n \ge 2$的无限族，其中${\rm PSp}(2n,q^2)$是自同构群.最后给出了一些应用实例。
英文摘要：
Let ${\cal Q}^-(2n+1,q)$ be an elliptic quadric of ${\rm PG}(2n+1,q)$. A relative $m$-ovoid of ${\cal Q}^-(2n+1,q)$ (with respect to a parablic section ${\cal Q} := {\cal Q}(2n,q) \subset {\cal Q}^-(2n+1,q)$) is a subset $\cal R$ of points of ${\cal Q}^-(2n+1,q)\setminus {\cal Q}$ such that every generator of ${\cal Q}^-(2n+1,q)$ not contained in $\cal Q$ meets $\cal R$ in precisely $m$ points. A relative $m$-ovoid having the same size as its complement (in ${\cal Q}^-(2n+1,q) \setminus {\cal Q}$) is called a relative hemisystem. We show that a nontrivial relative $m$-ovoid of ${\cal Q}^-(2n+1,q)$ is necessarily a relative hemisystem, forcing $q$ to be even. Also, we construct an infinite family of relative hemisystems of ${\cal Q}^-(4n+1,q)$, $n \ge 2$, admitting ${\rm PSp}(2n,q^2)$ as an automorphism group. Finally, some applications are given.
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