@UNPUBLISHED{B-Condition-Of-Intersecting-A-Projective-Variety-With-A-Varying-Linear-Subspace,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}
}

{"_id":"SQAWyvxJL4E3rRgJR","bibbaseid":"brgisser-conditionofintersectingaprojectivevarietywithavaryinglinearsubspace-2015","downloads":0,"creationDate":"2017-01-04T12:11:10.492Z","title":"Condition of intersecting a projective variety with a varying linear subspace","author_short":["Bürgisser, P."],"year":2015,"bibtype":"unpublished","biburl":"http://www3.math.tu-berlin.de/algebra/static/papers.parsed.bib","bibdata":{"bibtype":"unpublished","type":"unpublished","title":"Condition of intersecting a projective variety with a varying linear subspace","url":"http://arxiv.org/abs/1510.04142","author":[{"firstnames":["Peter"],"propositions":[],"lastnames":["Bürgisser"],"suffixes":[]}],"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 $ąppa_Z(L,z)$ 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 $vol(\\Sigma(Z))/vol(\\mathbb G(\\mathbb P^n,s)) = \\pi^{-1} (s+1)(n-s) \\deg(\\Sigma(Z))$.","month":"10","bibtex":"@UNPUBLISHED{B-Condition-Of-Intersecting-A-Projective-Variety-With-A-Varying-Linear-Subspace,\r\n TITLE={Condition of intersecting a projective variety with a varying linear subspace},\r\n URL={http://arxiv.org/abs/1510.04142},\r\n AUTHOR={Peter Bürgisser},\r\n YEAR={2015},\r\n 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 $vol(\\Sigma(Z))/vol(\\mathbb G(\\mathbb P^n,s)) = \\pi^{-1} (s+1)(n-s) \\deg(\\Sigma(Z))$.},\r\n MONTH={10}\r\n}\r\n\r\n","author_short":["Bürgisser, P."],"key":"B-Condition-Of-Intersecting-A-Projective-Variety-With-A-Varying-Linear-Subspace","id":"B-Condition-Of-Intersecting-A-Projective-Variety-With-A-Varying-Linear-Subspace","bibbaseid":"brgisser-conditionofintersectingaprojectivevarietywithavaryinglinearsubspace-2015","role":"author","urls":{"Paper":"http://arxiv.org/abs/1510.04142"},"downloads":0,"html":""},"search_terms":["condition","intersecting","projective","variety","varying","linear","subspace","bürgisser"],"keywords":[],"authorIDs":[],"dataSources":["mFLw2PzriqAWWFuqX"]}