Condition of intersecting a projective variety with a varying linear subspace

Abstract

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 $κ_Z(L,z)$ 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 $κ_Z(L,z) = 1/\sinα$, where $α$ is the minimum angle between the tangent spaces $T_zZ$ and $T_zL$. From this, we derive a condition number theorem that expresses $1/κ_Z(L,z)$ 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 $κ_Z(L) := \max κ_Z(L,z_i)$, taken over all intersection points $z_i\in Z\cap L$, leads to the study of the volume of tubes around the Hurwitz hypersurface $Σ(Z)$. As a first step towards this, we express the volume of $Σ(Z)$ in terms of its degree.