论文标题
$ {\ cal w}（5,2）$的$ M $ -OVOIDS $
The $m$-ovoids of ${\cal W}(5,2)$
论文作者
论文摘要
在本文中，我们关注的是$ m $ m $卵形，$ {\ cal w}（\ cal w}（2n+1，q）$，$ q $偶数。特别是，我们表明存在$ {\ rm pg}（2n+1，q）的椭圆形四，$不两极化至$ {\ cal w}（2n+1，q）$形成$ \ left（\ frac {q^n-1} {q^n-1} {q-1} {q-1} {q-1} {q-1} {q-1} {q-1} \ right）$ - oviot of $ ca;展出了另外一类$（q+1）$ - 卵形的$ {\ cal w}（5，q）$。它是通过将$ {\ rm psp}子组的两个子组粘合在一起而产生的。我们还表明，所获得的$ M $卵形并不属于迄今为止文献中已知的任何例子。此外，获取了$ {\ cal W}（5，2）$的$ M $ ovoids的计算机分类。事实证明，$ {\ cal w}（5，2）$具有$ m $ ovoids，并且仅当$ m = 3 $，并且恰好有三个成对的非晶状体示例。第一个示例来自椭圆四元$ {\ cal q}^ - （5，2）$偏振到$ {\ cal w}（5，2）$，而另外两个是前面提到的$ 3 $ ovoids。
In this paper we are concerned with $m$-ovoids of the symplectic polar space ${\cal W}(2n+1, q)$, $q$ even. In particular we show the existence of an elliptic quadric of ${\rm PG}(2n+1, q)$ not polarizing to ${\cal W}(2n+1, q)$ forming a $\left(\frac{q^n-1}{q-1}\right)$-ovoid of ${\cal W}(2n+1, q)$. A further class of $(q+1)$-ovoids of ${\cal W}(5, q)$ is exhibited. It arises by glueing together two orbits of a subgroup of ${\rm PSp}(6, q)$ isomorphic to ${\rm PSL}(2, q^2)$. We also show that the obtained $m$-ovoids do not fall in any of the examples known so far in the literature. Moreover, a computer classification of the $m$-ovoids of ${\cal W}(5, 2)$ is acquired. It turns out that ${\cal W}(5, 2)$ has $m$-ovoids if and only if $m = 3$ and that there are exactly three pairwise non-isomorphic examples. The first example comes from an elliptic quadric ${\cal Q}^-(5, 2)$ polarizing to ${\cal W}(5, 2)$, whereas the other two are the $3$-ovoids previously mentioned.
