Kyoto Journal of Mathematics

Strongly symmetric smooth toric varieties

M. Cuntz, Y. Ren, and G. Trautmann

Full-text: Open access

Abstract

We investigate toric varieties defined by arrangements of hyperplanes and call them strongly symmetric. The smoothness of such a toric variety translates to the fact that the arrangement is crystallographic. As a result, we obtain a complete classification of this class of toric varieties. Further, we show that these varieties are projective and describe associated toric arrangements in these varieties.

Article information

Source
Kyoto J. Math., Volume 52, Number 3 (2012), 597-620.

Dates
First available in Project Euclid: 26 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1343309708

Digital Object Identifier
doi:10.1215/21562261-1625208

Mathematical Reviews number (MathSciNet)
MR2959949

Zentralblatt MATH identifier
1270.14024

Subjects
Primary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15] 52C35: Arrangements of points, flats, hyperplanes [See also 32S22] 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx] 14M25: Toric varieties, Newton polyhedra [See also 52B20]

Citation

Cuntz, M.; Ren, Y.; Trautmann, G. Strongly symmetric smooth toric varieties. Kyoto J. Math. 52 (2012), no. 3, 597--620. doi:10.1215/21562261-1625208. https://projecteuclid.org/euclid.kjm/1343309708


Export citation

References

  • [AHS] N. Andruskiewitsch, I. Heckenberger, and H.-J. Schneider, The Nichols algebra of a semisimple Yetter-Drinfeld module, Amer. J. Math. 132 (2010), 1493–1547.
  • [BC] M. Barakat and M. Cuntz, Coxeter and crystallographic arrangements are inductively free, Adv. Math. 229 (2012), 691–709.
  • [BB1] V. Batyrev and M. Blume, On generalisations of Losev-Manin moduli spaces for classical root systems, Pure Appl. Math. Q. 7 (2011), 1053–1084.
  • [BB2] V. Batyrev and M. Blume, The functor of toric varieties associated with Weyl chambers and Losev-Manin moduli spaces, Tohoku Math. J. (2) 63 (2011), 581–604.
  • [BJ] M. Brion and R. Joshua, Equivariant Chow ring and Chern classes of wonderful symmetric varieties of minimal rank, Transform. Groups 13 (2008), 471–493.
  • [CH1] M. Cuntz and I. Heckenberger, Weyl groupoids of rank two and continued fractions, Algebra Number Theory 3 (2009), 317–340.
  • [CH2] M. Cuntz and I. Heckenberger, Weyl groupoids with at most three objects, J. Pure Appl. Algebra 213 (2009), 1112–1128.
  • [CH3] M. Cuntz and I. Heckenberger, Reflection groupoids of rank two and cluster algebras of type ${A}$, J. Combin. Theory Ser. A 118 (2011), 1350–1363.
  • [CH4] M. Cuntz and I. Heckenberger, Finite Weyl groupoids of rank three, Trans. Amer. Math. Soc. 364 (2012), 1316–1393.
  • [CH5] M. Cuntz and I. Heckenberger, Finite Weyl groupoids, preprint, arXiv:1008.5291v1 [math.CO]
  • [CLS] D. A. Cox, J. B. Little, and H. K. Schenck, Toric Varieties, Grad. Stud. Math. 124, Amer. Math. Soc., Providence, 2011.
  • [Cu] M. Cuntz, Crystallographic arrangements: Weyl groupoids and simplicial arrangements, Bull. Lond. Math. Soc. 43 (2011), 734–744.
  • [DCP1] C. De Concini and C. Procesi, “Complete symmetric varieties” in Invariant Theory (Montecatini, 1982), Lecture Notes in Math. 996, Springer, Berlin, 1983, 1–44.
  • [DCP2] C. De Concini and C. Procesi, On the geometry of toric arrangements, Transform. Groups 10 (2005), 387–422.
  • [DL] I. Dolgachev and V. Lunts, A character formula for the representation of a Weyl group in the cohomology of the associated toric variety, J. Algebra 168 (1994), 741–772.
  • [Grü] B. Grünbaum, A catalogue of simplicial arrangements in the real projective plane, Ars Math. Contemp. 2 (2009), 1–25.
  • [Hec] I. Heckenberger, The Weyl groupoid of a Nichols algebra of diagonal type, Invent. Math. 164 (2006), 175–188.
  • [HS] I. Heckenberger and H.-J. Schneider, Root systems and Weyl groupoids for Nichols algebras, Proc. Lond. Math. Soc. (3) 101 (2010), 623–654.
  • [HW] I. Heckenberger and V. Welker, Geometric combinatorics of Weyl groupoids, J. Algebraic Combin. 34 (2011), 115–139.
  • [Kly] A. A. Klyachko, Toric varieties and flag spacies (in Russian), Tr. Mat. Inst. Steklova 208 (1995), 139–162.
  • [Oda] T. Oda, Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties, Ergeb. Math. Grenzgeb. (3) 15, Springer, Berlin, 1988.
  • [OT] P. Orlik and H. Terao, Arrangements of Hyperplanes, Grundlehren Math. Wiss. 300, Springer, Berlin, 1992.
  • [Pro] C. Procesi, “The toric variety associated to Weyl chambers” in Mots, Lang. Raison. Calc., Hermès, Paris, 1990, 153–161.
  • [Ste] J. R. Stembridge, Some permutation representations of Weyl groups associated with the cohomology of toric varieties, Adv. Math. 106 (1994), 244–301.
  • [VK] V. E. Voskresenskij and A. A. Klyachko, Toroidal Fano varieties and root systems, Math. USSR Izv. 24 (1985), 221–244.