Condition of intersecting a projective variety with a varying linear subspace

TL;DR

This paper investigates the numerical stability of intersecting a complex projective variety with a varying linear subspace, defining a condition number related to tangent space angles and analyzing its probabilistic behavior.

Contribution

It introduces a new intersection condition number based on tangent space angles and relates it to the distance from ill-posed intersections, with a probabilistic analysis of its maximum value.

Findings

The condition number is inversely proportional to the sine of the minimal angle between tangent spaces.

A condition number theorem links the inverse of the condition number to the distance from the local Schubert variety.

The volume of the Hurwitz hypersurface is expressed in terms of its degree.

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 $\kappa_Z(L,z)$ 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.