### Abstract

We define the flag space and space of singular vectors for an arrangement A of hyperplanes in projective space equipped with a system of weights a: A → ℂ. We show that the contravariant bilinear form of the corresponding weighted central arrangement induces a well-defined form on the space of singular vectors of the projectivization. If Σ_{HεA} a(H) = 0, this form is naturally isomorphic to the restriction to the space of singular vectors of the contravariant form of any affine arrangement obtained from A by dehomogenizing with respect to one of its hyperplanes.

Original language | English (US) |
---|---|

Title of host publication | Configuration Spaces |

Subtitle of host publication | Geometry, Combinatorics and Topology |

Publisher | Scuola Normale Superiore |

Pages | 255-272 |

Number of pages | 18 |

ISBN (Electronic) | 9788876424311 |

ISBN (Print) | 9788876424304 |

State | Published - Jan 1 2012 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Configuration Spaces: Geometry, Combinatorics and Topology*(pp. 255-272). Scuola Normale Superiore.

**The contravariant form on singular vectors of a projective arrangement.** / Falk, Michael J; Varchenko, Alexander N.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Configuration Spaces: Geometry, Combinatorics and Topology.*Scuola Normale Superiore, pp. 255-272.

}

TY - CHAP

T1 - The contravariant form on singular vectors of a projective arrangement

AU - Falk, Michael J

AU - Varchenko, Alexander N.

PY - 2012/1/1

Y1 - 2012/1/1

N2 - We define the flag space and space of singular vectors for an arrangement A of hyperplanes in projective space equipped with a system of weights a: A → ℂ. We show that the contravariant bilinear form of the corresponding weighted central arrangement induces a well-defined form on the space of singular vectors of the projectivization. If ΣHεA a(H) = 0, this form is naturally isomorphic to the restriction to the space of singular vectors of the contravariant form of any affine arrangement obtained from A by dehomogenizing with respect to one of its hyperplanes.

AB - We define the flag space and space of singular vectors for an arrangement A of hyperplanes in projective space equipped with a system of weights a: A → ℂ. We show that the contravariant bilinear form of the corresponding weighted central arrangement induces a well-defined form on the space of singular vectors of the projectivization. If ΣHεA a(H) = 0, this form is naturally isomorphic to the restriction to the space of singular vectors of the contravariant form of any affine arrangement obtained from A by dehomogenizing with respect to one of its hyperplanes.

UR - http://www.scopus.com/inward/record.url?scp=85027810308&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85027810308&partnerID=8YFLogxK

M3 - Chapter

SN - 9788876424304

SP - 255

EP - 272

BT - Configuration Spaces

PB - Scuola Normale Superiore

ER -