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Condition of intersecting a projective variety with a varying linear subspace
Zitatschlüssel B-Condition-Of-Intersecting-A-Projective-Variety-With-A-Varying-Linear-Subspace
Autor Peter Bürgisser
Jahr 2015
Monat 10
Notiz To appear in SIAGA.
Zusammenfassung 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 α is the minimum angle between the tangent spaces $T_zZ$ and $T_zL$. From this, we derive a condition number theorem that expresses $1/\kappa_Z(L,z)$ 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 $vol(\Sigma(Z))/vol(\mathbb G(\mathbb P^n,s)) = \pi^-1 (s+1)(n-s) \deg(\Sigma(Z))$.
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