论文标题
$ \ mathbb {rp}^n $中不同交叉点的投影线的所有三元空间
The space of all triples of projective lines of distinct intersections in $\mathbb{RP}^n$
论文作者
论文摘要
我们研究了$ \ mathbb {rp}^n $中所有射击线的三元组的空间，以使三线中的任何线都在不同的点上与另外两个线相交。我们表明，对于$ n = 2 $和$ 3 $，这些空间在同质上等同于实际的完整标志品种$ flag（\ Mathbb {r}^n）$，分别为$ n = 3 $和$ 4，$，我们明确地计算了相应空间的整体同源性。我们证明，对于任意$ n $，此空间是同等的，相当于$ flag（1,2,3，\ mathbb {r}^{n+1}），$ signature $（0,1,2,3，n+1）$的所有部分标志在$（n+1）$（n+1）$（n+1）$ - $ - $ 1）$ - $ - $ - $ - $ \ mathbb $ \ mathbb {
We study the space of all triples of projective lines in $\mathbb{RP}^n$ such that any line in a triple intersects the two others at distinct points. We show that for $n=2$ and $3$ these spaces are homotopically equivalent to the real complete flag variety $Flag(\mathbb{R}^n)$ for $n=3$ and $4,$ respectively, and we explicitly calculate the integral homology of the corresponding spaces. We prove that for arbitrary $n$, this space is homotopy equivalent to $Flag(1,2,3,\mathbb{R}^{n+1}),$ the variety of all partial flags of signature $(0,1,2,3,n+1)$ in an $(n+1)$-dimensional vector space over $\mathbb{R}.$
