Discriminantal Arrangements, Fiber Polytopes and Formality
Margaret M. Bayer
and Keith A. Brandt
DOI: 10.1023/A:1008601810383
Abstract
Manin and Schechtman defined the discriminantal arrangement of a generic hyperplane arrangement as a generalization of the braid arrangement. This paper shows their construction is dual to the fiber zonotope construction of Billera and Sturmfels, and thus makes sense even when the base arrangement is not generic. The hyperplanes, face lattices and intersection lattices of discriminantal arrangements are studied. The discriminantal arrangement over a generic arrangement is shown to be formal (and in some cases 3-formal), though it is in general not free. An example of a free discriminantal arrangement over a generic arrangement is given.
Pages: 229–246
Keywords: discriminantal arrangement; hyperplane arrangement; polytope; free
Full Text: PDF
References
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2. L. Billera and B. Sturmfels, “Fiber polytopes,” Ann. of Math. 135 (1992), 527-549.
3. A. Bj\ddot orner, M. Las Vergnas, B. Sturmfels, N. White and G. Ziegler, “Oriented Matroids,” Encyclopedia of Mathematics and Its Applications 46, Cambridge University Press, Cambridge, 1993.
4. K. Brandt and H. Terao, “Free arrangements and relation spaces,” Disc. & Comp. Geometry 12 (1994), 49-63.
5. P. Edelman and V. Reiner, “Free arrangements and rhombic tilings,” Disc. & Comp. Geometry 15 (1996), 307-340.
6. M. Falk, “A note on discriminantal arrangements,” Proc. Amer. Math. Soc. 122 (1994), 1221-1227.
7. M. Falk and R. Randell, “On the homotopy theory of arrangements,” In: Complex Analytic Singularities, Advanced Studies in Pure Math. 8 (1986), 101-124.
8. Y.I. Manin and V.V. Schechtman, “Arrangements of hyperplanes, higher braid groups and higher Bruhat orders,” In: Algebraic Number Theory-in honor of K. Iwasawa, Advanced Studies in Pure Math. 17 (1989), 289-308.
9. P. Orlik and H. Terao, “Arrangements of Hyperplanes,” Grundlehren der Mathematischen Wissenschaften 300, Springer-Verlag, 1992.
10. S. Yuzvinsky, “First two obstructions to the freeness of arrangements,” Trans. Amer. Math. Soc. 335 (1993), 231-244.
11. G.M. Ziegler, “Higher Bruhat orders and cyclic hyperplane arrangements,” Topology 32 (1993), 259-279.