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The first two obstructions to the freeness of arrangements


Author: Sergey Yuzvinsky
Journal: Trans. Amer. Math. Soc. 335 (1993), 231-244
MSC: Primary 52B30; Secondary 32S20
DOI: https://doi.org/10.1090/S0002-9947-1993-1089421-5
MathSciNet review: 1089421
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Abstract: In his previous paper the author characterized free arrangements by the vanishing of cohomology modules of a certain sheaf of graded modules over a polynomial ring. Thus the homogeneous components of these cohomology modules can be viewed as obstructions to the freeness of an arrangement. In this paper the first two obstructions are studied in detail. In particular the component of degree zero of the first nontrivial cohomology module has a close relation to formal arrangements and to the operation of truncation. This enables us to prove that in dimension greater than two every free arrangement is formal and not a proper truncation of an essential arrangement.


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  • [FR] M. Falk and R. Randell, On the homotopy theory of arrangements, Adv. Stud. Pure Math. 8 (1986), 101-124. MR 894288 (88f:32045)
  • [G] R. Godement, Topologie algébrique et théorie des faisceaux, Hermann, Paris, 1958. MR 0102797 (21:1583)
  • [ST] L. Solomon and H. Terao, A formula for the characteristic polynomial of an arrangement, Adv. in Math. 64 (1987), 305-325. MR 888631 (88m:32022)
  • [T1] H. Terao, Arrangements of hyperplanes and their freeness. I, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27(2) (1980), 293-312. MR 586451 (84i:32016a)
  • [T2] -, Generalized exponents of a free arrangements of hyperplanes and Shephard-Todd-Brieskorn formula, Invent. Math. 63 (1981), 159-179. MR 608532 (82e:32018b)
  • [T3] -, Free arrangements of hyperplanes over an arbitrary field, Proc. Japan Acad. 59 (1987), 301-304. MR 726186 (85f:32017)
  • [Y] S. Yuzvinsky, Cohomology of local sheaves on arrangement lattices, Proc. Amer. Math. Soc. 112 (1991), 1207-1217. MR 1062840 (91j:52016)
  • [Z] G. Ziegler, Combinatorial construction of logarithmic differential forms, Adv. in Math. 76 (1989), 116-154. MR 1004488 (90j:32016)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1089421-5
Article copyright: © Copyright 1993 American Mathematical Society

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