Critical Points and Resonance of Hyperplane Arrangements

Cohen, D. ; Denham, G. ; Falk, M. ; Varchenko, A.

Canadian journal of mathematics, Tome 63 (2011) no. 5, pp. 1038-1057

Résumé

If ${{\Phi }_{\lambda }}$ is a master function corresponding to a hyperplane arrangement $\mathcal{A}$ and a collection of weights $\lambda $ , we investigate the relationship between the critical set of ${{\Phi }_{\lambda }}$ , the variety defined by the vanishing of the one-form ${{\omega }_{\lambda }}=\text{d}\log {{\Phi }_{\lambda }}$ , and the resonance of $\lambda $ . For arrangements satisfying certain conditions, we show that if $\lambda $ is resonant in dimension $p$ , then the critical set of ${{\Phi }_{\lambda }}$ has codimension at most $p$ . These include all free arrangements and all rank 3 arrangements.

Détail

DOI : 10.4153/CJM-2011-028-8

Mots-clés : 32S22, 55N25, 52C35, hyperplane arrangement, master function, resonant weights, critical set

Bibliographie

[Coh02] [Coh02] Cohen, D., Triples of arrangements and local systems. Proc. Amer. Math. Soc. 130(2002), no. 10, 3025–3031. doi:10.1090/S0002-9939-02-06428-6 Google Scholar

[CV03] [CV03] Cohen, D. and A. N. Varchenko, Resonant local systems on complements of discriminantal arrangements and sl 2 representations. Geom. Dedicata 101(2003), 217–233. doi:10.1023/A:1026370732724 Google Scholar

[Dam99] [Dam99] Damon, J., Critical points of affine multiforms on the complements of arrangements. In: Singularity theory (Liverpool, 1996), London Math. Soc. Lecture Note Ser., 263, Cambridge University Press, Cambridge, 1999, pp. 25–53. Google Scholar

[Den07] [Den07] Denham, G., Zeroes of 1-forms and resonance of free arrangements, Oberwolfach Rep. 4 (2007), no. 3, 2345–2347. Google Scholar

[Dim08] [Dim08] Dimca, A., Characteristic varieties and logarithmic differential 1-forms. Compos. Math. 146(2010), no. 1, 129–144. doi:10.1112/S0010437X09004461 Google Scholar

[Eis95] [Eis95] Eisenbud, D., Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995. Google Scholar

[ESV92] [ESV92] Esnault, H., V. Schechtman, and V. Viehweg, Cohomology of local systems on the complement of hyperplanes. Invent. Math. 109(1992), no. 3, 557–561. Erratum, ibid., 112(1993), no. 2, 447. doi:10.1007/BF01232040 Google Scholar

[Fal07] [Fal07] Falk, M., Resonance and zeros of logarithmic one-forms with hyperplane poles, Oberwolfach Rep. 4 (2007), no. 3, 2343–2345. Google Scholar

[FY07] [FY07] Falk, M. and S. Yuzvinsky, Multinets, resonance varieties, and pencils of plane curves. Compos. Math. 143(2007), no. 4, 1069–1088. Google Scholar

[Far01] [Far01] Farber, M., Topology of closed 1-forms and their critical points. Topology 40(2001), no. 2, 235–258. doi:10.1016/S0040-9383(99)00059-2 Google Scholar

[Far04] [Far04] Farber, M., Topology of closed one-forms. Mathematical Surveys and Monographs, 108, American Mathematical Society, Providence, RI, 2004. Google Scholar

[GS] [GS] Grayson, D. and M. Stillman, Macaulay 2: a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2. Google Scholar

[HKS05] [HKS05] Ho, S.şten, A. Khetan, and B. Sturmfels, Solving the likelihood equations. Found. Comput. Math. 5(2005), no. 4, 389–407. doi:10.1007/s10208-004-0156-8 Google Scholar

[MV04] [MV04] Mukhin, E. and A. Varchenko, Critical points of master functions and flag varieties. Commun. Contemp. Math. 6(2004), no. 1, 111–163. doi:10.1142/S0219199704001288 Google Scholar

[MV05] [MV05] Mukhin, E. and A. Varchenko, Miura opers and critical points of master functions. Cent. Eur. J. Math. 3(2005), no. 2, 155–182 (electronic). doi:10.2478/BF02479193 Google Scholar

[MS01] [MS01] Musta, M.ţă and H. Schenck, The module of logarithmic p-forms of a locally free arrangement. J. Algebra 241(2001), no. 2, 699–719. doi:10.1006/jabr.2000.8606 Google Scholar

[OS02] [OS02] Orlik, P. and R. Silvotti, Local system homology of arrangement complements. In: Arrangements—Tokyo 1998, Adv. Stud. Pure Math., 27, Kinokuniya, Tokyo, 2000, pp. 247–256. Google Scholar

[OT92] [OT92] Orlik, P. and H. Terao, Arrangements of hyperplanes. Grundlehren Mathematischen Wissenschaften, 300, Springer-Verlag, Berlin, 1992. Google Scholar

[OT95a] [OT95a] Orlik, P. and H. Terao, The number of critical points of a product of powers of linear functions. Invent. Math. 120(1995), no. 1, 1–14. doi:10.1007/BF01241120 Google Scholar

[OT95b] [OT95b] Orlik, P. and H. Terao, Arrangements and Milnor fibers. Math. Ann. 301(1995), no. 2, 211–235. doi:10.1007/BF01446627 Google Scholar

[OT01] [OT01] Orlik, P. and H. Terao, Arrangements and hypergeometric integrals. MSJ Memoirs, 9, Mathematical Society of Japan, Tokyo, 2001. Google Scholar

[RV95] [RV95] Reshetikhin, N. and A. Varchenko, Quasiclassical asymptotics of solutions to the KZ equations. In: Geometry, topology, and physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA, 1995, pp. 293–322. Google Scholar

[STV95] [STV95] Schechtman, V., H. Terao, and A. Varchenko, Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors. J. Pure Appl. Algebra 100(1995), no. 1–3, 93–102. doi:10.1016/0022-4049(95)00014-N Google Scholar

[SV91] [SV91] Schechtman, V. and A. Varchenko, Arrangements of hyperplanes and Lie algebra homology. Invent. Math. 106(1991), no. 1, 139–194. doi:10.1007/BF01243909 Google Scholar

[SV03] [SV03] Scherbak, I. and A. Varchenko, Critical points of functions, sl representations, and Fuchsian differential equations with only univalued solutions. Mosc. Math. J. 3(2003), no. 2, 621–645, 745. 2 Google Scholar

[Sil96] [Sil96] Silvotti, R., On a conjecture of Varchenko. Invent. Math. 126(1996), no. 2, 235–248. doi:10.1007/s002220050096 Google Scholar

[Suc02] [Suc02] I, A.. Suciu, Translated tori in the characteristic varieties of complex hyperplane arrangements. Topology Appl. 118(2002), no. 1–2, 209–223. doi:10.1016/S0166-8641(01)00052-9 Google Scholar

[TY95] [TY95] Terao, H. and S. Yuzvinsky, Logarithmic forms on affine arrangements. Nagoya Math. J. 139(1995), 129–149. Google Scholar

[Var95] [Var95] Varchenko, A., Critical points of the product of powers of linear functions and families of bases of singular vectors. Compositio Math. 97(1995), no. 3, 385–401. Google Scholar

[Var06] [Var06] Varchenko, A., Bethe ansatz for arrangements of hyperplanes and the Gaudin model. Mosc. Math. J. 6(2006), no. 1, 195–210, 223–224. Google Scholar

[WY97] [WY97] Wiens, J. and S. Yuzvinsky, De Rham cohomology of logarithmic forms on arrangements of hyperplanes. Trans. Amer. Math. Soc. 349(1997), no. 4, 1653–1662. doi:10.1090/S0002-9947-97-01894-1 Google Scholar

[Yuz95] [Yuz95] Yuzvinsky, S., Cohomology of the Brieskorn-Orlik-Solomon algebras. Comm. Algebra 23(1995), no. 14, 5339–5354. doi:10.1080/00927879508825535 Google Scholar

[Suc02] [Suc02] I, A.. Suciu, Translated tori in the characteristic varieties of complex hyperplane arrangements. Topology Appl. 118(2002), no. 1–2, 209–223. doi:10.1016/S0166-8641(01)00052-9 Google Scholar

[TY95] [TY95] Terao, H. and S. Yuzvinsky, Logarithmic forms on affine arrangements. Nagoya Math. J. 139(1995), 129–149. Google Scholar

[Var95] [Var95] Varchenko, A., Critical points of the product of powers of linear functions and families of bases of singular vectors. Compositio Math. 97(1995), no. 3, 385–401. Google Scholar

[Var06] [Var06] Varchenko, A., Bethe ansatz for arrangements of hyperplanes and the Gaudin model. Mosc. Math. J. 6(2006), no. 1, 195–210, 223–224. Google Scholar

[WY97] [WY97] Wiens, J. and S. Yuzvinsky, De Rham cohomology of logarithmic forms on arrangements of hyperplanes. Trans. Amer. Math. Soc. 349(1997), no. 4, 1653–1662. doi:10.1090/S0002-9947-97-01894-1 Google Scholar

[Yuz95] [Yuz95] Yuzvinsky, S., Cohomology of the Brieskorn-Orlik-Solomon algebras. Comm. Algebra 23(1995), no. 14, 5339–5354. doi:10.1080/00927879508825535 Google Scholar