Free and locally free arrangements with a given intersection lattice

Author:
Sergey Yuzvinsky

Journal:
Proc. Amer. Math. Soc. **118** (1993), 745-752

MSC:
Primary 52B30; Secondary 05B35, 06A09, 32S20

MathSciNet review:
1160307

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In previous papers the author characterized free arrangements of hyperplanes by the vanishing of cohomology of the intersection lattice with coefficients in a certain sheaf of graded modules over a polynomial ring. The main result of this paper is that for a locally free arrangement the degrees of nonzero homogeneous components of the cohomology modules are bounded by a number depending only on the intersection lattice. In particular, the Hilbert coefficients of the module of derivations of a locally free arrangement are combinatorial invariants. Another result of the paper asserts that the set of free arrangements is Zariski open in the set of all arrangements with a given intersection lattice.

**[G]**Roger Godement,*Topologie algébrique et théorie des faisceaux*, Actualit’es Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13, Hermann, Paris, 1958 (French). MR**0102797****[M]**Hideyuki Matsumura,*Commutative algebra*, W. A. Benjamin, Inc., New York, 1970. MR**0266911****[RT]**Lauren L. Rose and Hiroaki Terao,*Hilbert polynomials and geometric lattices*, Adv. Math.**84**(1990), no. 2, 209–225. MR**1080977**, 10.1016/0001-8708(90)90045-O**[T1]**Hiroaki Terao,*Arrangements of hyperplanes and their freeness. I*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**27**(1980), no. 2, 293–312. MR**586451****[T2]**Hiroaki Terao,*Generalized exponents of a free arrangement of hyperplanes and Shepherd-Todd-Brieskorn formula*, Invent. Math.**63**(1981), no. 1, 159–179. MR**608532**, 10.1007/BF01389197**[T3]**Hiroaki Terao,*Free arrangements of hyperplanes over an arbitrary field*, Proc. Japan Acad. Ser. A Math. Sci.**59**(1983), no. 7, 301–303. MR**726186****[Y1]**Sergey Yuzvinsky,*Cohomology of local sheaves on arrangement lattices*, Proc. Amer. Math. Soc.**112**(1991), no. 4, 1207–1217. MR**1062840**, 10.1090/S0002-9939-1991-1062840-2**[Y2]**Sergey Yuzvinsky,*The first two obstructions to the freeness of arrangements*, Trans. Amer. Math. Soc.**335**(1993), no. 1, 231–244. MR**1089421**, 10.1090/S0002-9947-1993-1089421-5

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
52B30,
05B35,
06A09,
32S20

Retrieve articles in all journals with MSC: 52B30, 05B35, 06A09, 32S20

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1993-1160307-6

Article copyright:
© Copyright 1993
American Mathematical Society