TY - CHAP

T1 - Geometry and combinatorics of resonant weights

AU - Falk, Michael

N1 - Publisher Copyright:
© 2009, Birkhäuser Verlag Basel/Switzerland.

PY - 2010

Y1 - 2010

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

AN - SCOPUS:85028752830

T3 - Progress in Mathematics

SP - 155

EP - 176

BT - Progress in Mathematics

PB - Springer Basel

ER -