### Abstract

Let A be an arrangement of n hyperplanes in ℂ^{ℓ}. Let k be a field and A=⊕_{p=0} ^{ℓ} A ^{p} the Orlik-Solomon algebra of A over k. The p ^{th} resonance variety of A over k is the set R^{p} (A, k) of one-forms a∈A ^{1} annihilated by some b∈A ^{p}\(a). This notion arises naturally from consideration of cohomology of the rank-one local systems generated by single-valued branches of A-master functions Φ_{a}. For the most part we focus on the case p=1. We will describe the features of R ^{1} (A, k) for k = C and also for fields of positive characteristic, and their connections with other phenomena. We derive simple necessary and sufficient conditions for an element a to lie in R ^{1} (A, k), and consequently obtain a precise description of R ^{1} (A, k) as a ruled variety. We sketch the description of components of R ^{1} (A, C) in terms of multinets, and the related Ceva-type pencils of plane curves. We present examples over fields of positive characteristic showing that the ruling may be quite nontrivial. In particular, R ^{1} (A, k) need not be a union of linear varieties, in contrast to the characteristic-zero case. We also give an example for which components of R ^{1} (A, k) do not intersect trivially. We discuss the current state of the classification problem for Orlik-Solomon algebras, and the utility of resonance varieties in this context. Finally we sketch a relationship between one-forms a ∈ R ^{p} (A, k) and the critical loci of the corresponding master functions Φ_{a}. For p=1 we obtain a precise connection using the associated multinet and Ceva-type pencil.

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

Title of host publication | Progress in Mathematics |

Publisher | Springer Basel |

Pages | 155-176 |

Number of pages | 22 |

Volume | 283 |

DOIs | |

State | Published - Jan 1 2010 |

### Publication series

Name | Progress in Mathematics |
---|---|

Volume | 283 |

ISSN (Print) | 0743-1643 |

ISSN (Electronic) | 2296-505X |

### Fingerprint

### Keywords

- Arrangement
- Local system cohomology
- Master function
- Multinet
- Net
- Orlik-Solomon algebra
- Pencil
- Resonance variety

### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Geometry and Topology

### Cite this

*Progress in Mathematics*(Vol. 283, pp. 155-176). (Progress in Mathematics; Vol. 283). Springer Basel. https://doi.org/10.1007/978-3-0346-0209-9_6

**Geometry and combinatorics of resonant weights.** / Falk, Michael J.

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

*Progress in Mathematics.*vol. 283, Progress in Mathematics, vol. 283, Springer Basel, pp. 155-176. https://doi.org/10.1007/978-3-0346-0209-9_6

}

TY - CHAP

T1 - Geometry and combinatorics of resonant weights

AU - Falk, Michael J

PY - 2010/1/1

Y1 - 2010/1/1

N2 - Let A be an arrangement of n hyperplanes in ℂℓ. Let k be a field and A=⊕p=0 ℓ A p the Orlik-Solomon algebra of A over k. The p th resonance variety of A over k is the set Rp (A, k) of one-forms a∈A 1 annihilated by some b∈A p\(a). This notion arises naturally from consideration of cohomology of the rank-one local systems generated by single-valued branches of A-master functions Φa. For the most part we focus on the case p=1. We will describe the features of R 1 (A, k) for k = C and also for fields of positive characteristic, and their connections with other phenomena. We derive simple necessary and sufficient conditions for an element a to lie in R 1 (A, k), and consequently obtain a precise description of R 1 (A, k) as a ruled variety. We sketch the description of components of R 1 (A, C) in terms of multinets, and the related Ceva-type pencils of plane curves. We present examples over fields of positive characteristic showing that the ruling may be quite nontrivial. In particular, R 1 (A, k) need not be a union of linear varieties, in contrast to the characteristic-zero case. We also give an example for which components of R 1 (A, k) do not intersect trivially. We discuss the current state of the classification problem for Orlik-Solomon algebras, and the utility of resonance varieties in this context. Finally we sketch a relationship between one-forms a ∈ R p (A, k) and the critical loci of the corresponding master functions Φa. For p=1 we obtain a precise connection using the associated multinet and Ceva-type pencil.

AB - Let A be an arrangement of n hyperplanes in ℂℓ. Let k be a field and A=⊕p=0 ℓ A p the Orlik-Solomon algebra of A over k. The p th resonance variety of A over k is the set Rp (A, k) of one-forms a∈A 1 annihilated by some b∈A p\(a). This notion arises naturally from consideration of cohomology of the rank-one local systems generated by single-valued branches of A-master functions Φa. For the most part we focus on the case p=1. We will describe the features of R 1 (A, k) for k = C and also for fields of positive characteristic, and their connections with other phenomena. We derive simple necessary and sufficient conditions for an element a to lie in R 1 (A, k), and consequently obtain a precise description of R 1 (A, k) as a ruled variety. We sketch the description of components of R 1 (A, C) in terms of multinets, and the related Ceva-type pencils of plane curves. We present examples over fields of positive characteristic showing that the ruling may be quite nontrivial. In particular, R 1 (A, k) need not be a union of linear varieties, in contrast to the characteristic-zero case. We also give an example for which components of R 1 (A, k) do not intersect trivially. We discuss the current state of the classification problem for Orlik-Solomon algebras, and the utility of resonance varieties in this context. Finally we sketch a relationship between one-forms a ∈ R p (A, k) and the critical loci of the corresponding master functions Φa. For p=1 we obtain a precise connection using the associated multinet and Ceva-type pencil.

KW - Arrangement

KW - Local system cohomology

KW - Master function

KW - Multinet

KW - Net

KW - Orlik-Solomon algebra

KW - Pencil

KW - Resonance variety

UR - http://www.scopus.com/inward/record.url?scp=85028752830&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85028752830&partnerID=8YFLogxK

U2 - 10.1007/978-3-0346-0209-9_6

DO - 10.1007/978-3-0346-0209-9_6

M3 - Chapter

VL - 283

T3 - Progress in Mathematics

SP - 155

EP - 176

BT - Progress in Mathematics

PB - Springer Basel

ER -