Condition of Intersecting a Projective Variety with a Varying Linear Subspace
Condition of Intersecting a Projective Variety with a Varying Linear Subspace
复制标题
射影簇与变线性子空间的相交条件
DOI：
10.1137/16m1077222
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Peter Bürgisser
中科院分区：
文献类型：
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作者：
Peter Bürgisser
研究了固定m维不可约复射影簇Z\subseteq\mathbb{P}^n$与补维数s=n-m$的变线性子空间L\subseteq\mathbb{P}^n$相交问题的数值条件.我们定义在Z\cap L$中光滑交点$z\处的相交条件数$\kappa_Z（L，z）$作为局部定义的解映射$\mathbb{G}（s，\mathbb{P}^n）\到\mathbb{P}^n，\，L\mapsto z$的导数的范数。我们证明了$\kappa_Z（L，z）= 1/\sin\alpha$，其中$\alpha$是切空间$T_zZ$和$T_zL$之间的最小夹角。由此，我们导出了一个条件数定理，将$1/\kappa_Z（L，z）$表示为$L$到局部舒伯特簇的距离，局部舒伯特簇由与$Z$在$z$处有不适定交的线性子空间组成。通过对最大条件数$\kappa_Z（L）：= \max \kappa_Z（L，z_i）$的概率分析，得到了在Z\cap L$中所有交点$z_i\上的最大条件数，从而研究了Hurwitz超曲面$\Sigma（Z）$周围的管的体积。作为实现这一点的第一步，我们用它的度来表示$\Sigma（Z）$的体积。
The numerical condition of the problem of intersecting a fixed $m$-dimensional irreducible complex projective variety $Z\subseteq\mathbb{P}^n$ with a varying linear subspace $L\subseteq\mathbb{P}^n$ of complementary dimension $s=n-m$ is studied. We define the intersection condition number $\kappa_Z(L,z)$ at a smooth intersection point $z\in Z\cap L$ as the norm of the derivative of the locally defined solution map $\mathbb{G}(s,\mathbb{P}^n)\to\mathbb{P}^n,\, L\mapsto z$. We show that $\kappa_Z(L,z) = 1/\sin\alpha$, where $\alpha$ is the minimum angle between the tangent spaces $T_zZ$ and $T_zL$. From this, we derive a condition number theorem that expresses $1/\kappa_Z(L,z)$ as the distance of $L$ to the local Schubert variety, which consists of the linear subspaces having an ill-posed intersection with $Z$ at $z$. A probabilistic analysis of the maximum condition number $\kappa_Z(L) := \max \kappa_Z(L,z_i)$, taken over all intersection points $z_i\in Z\cap L$, leads to the study of the volume of tubes around the Hurwitz hypersurface $\Sigma(Z)$. As a first step towards this, we express the volume of $\Sigma(Z)$ in terms of its degree.
DOI：
10.1007/978-3-642-38896-5
发表时间：
2013-08
期刊：
--
影响因子：
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作者：
Peter Bürgisser;F. Cucker
通讯作者：
Peter Bürgisser;F. Cucker
影响因子：
3
作者：
D. Amelunxen;P. Bürgisser
通讯作者：
P. Bürgisser