10 2015.
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 $ąppa_Z(L,z)$ at a smooth intersection point $z∈Z∩L$ as the norm of the derivative of the locally defined solution map $\mathbb G(\mathbb P^n,s)\to\mathbb P^n,\, L↦z$. We show that $ąppa_Z(L,z) = 1/\sin\alpha$, 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 $1/ąppa_Z(L,z)$ 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 $ąppa_Z(L) := \max ąppa_Z(L,z_i)$, taken over all intersection points $z_i∈Z∩L$, leads to the study of the volume of tubes around the Hurwitz hypersurface $\Sigma(Z)$. As a first step towards this, we prove that $vol(\Sigma(Z))/vol(\mathbb G(\mathbb P^n,s)) = \pi^{-1} (s+1)(n-s) \deg(\Sigma(Z))$.
@UNPUBLISHED{B-Condition-Of-Intersecting-A-Projective-Variety-With-A-Varying-Linear-Subspace,
TITLE={Condition of intersecting a projective variety with a varying linear subspace},
URL={http://arxiv.org/abs/1510.04142},
AUTHOR={Peter Bürgisser},
YEAR={2015},
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 $\mathbb G(\mathbb P^n,s)\to\mathbb P^n,\, L\mapsto z$. 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 prove that $vol(\Sigma(Z))/vol(\mathbb G(\mathbb P^n,s)) = \pi^{-1} (s+1)(n-s) \deg(\Sigma(Z))$.},
MONTH={10}
}