Tohoku Mathematical Journal

The functor of toric varieties associated with Weyl chambers and Losev-Manin moduli spaces

Victor Batyrev and Mark Blume

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Abstract

A root system $R$ of rank $n$ defines an $n$-dimensional smooth projective toric variety $X(R)$ associated with its fan of Weyl chambers. We give a simple description of the functor of $X(R)$ in terms of the root system $R$ and apply this result in the case of root systems of type $A$ to give a new proof of the fact that the toric variety $X(A_n)$ is the fine moduli space $\overline{L}_{n+1}$ of stable $(n+1)$-pointed chains of projective lines investigated by Losev and Manin.

Article information

Source
Tohoku Math. J. (2), Volume 63, Number 4 (2011), 581-604.

Dates
First available in Project Euclid: 6 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1325886282

Digital Object Identifier
doi:10.2748/tmj/1325886282

Mathematical Reviews number (MathSciNet)
MR2872957

Zentralblatt MATH identifier
1255.14041

Subjects
Primary: 14M25: Toric varieties, Newton polyhedra [See also 52B20]
Secondary: 14D22: Fine and coarse moduli spaces 14H10: Families, moduli (algebraic)

Keywords
Toric varieties root systems Losev-Manin moduli spaces

Citation

Batyrev, Victor; Blume, Mark. The functor of toric varieties associated with Weyl chambers and Losev-Manin moduli spaces. Tohoku Math. J. (2) 63 (2011), no. 4, 581--604. doi:10.2748/tmj/1325886282. https://projecteuclid.org/euclid.tmj/1325886282


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References

  • A. Ash, D. Mumford, M. Rapoport and Y. Tai, Smooth compactification of locally symmetric varieties, Math. Sci. Press, Brookline, Mass., 1975.
  • V. Batyrev, On the classification of smooth projective toric varieties, Tohoku Math. J. 43 (1991), 569–585.
  • V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994), 493–535.
  • V. Batyrev and M. Blume, On generalisations of Losev-Manin moduli spaces for classical root systems, Pure Appl. Math. Q. 7 (2011), 1053–1084.
  • S. Bloch and D. Kreimer, Mixed Hodge structures and renormalization in physics, Commun. Number Theory Phys. 2 (2008), 637–718.
  • N. Bourbaki, Groupes et algèbres de Lie (Ch. 4–6), Hermann, Paris, 1968.
  • M. Brion and R. Joshua, Equivariant Chow ring and Chern classes of wonderful symmetric varieties of minimal rank, Transform. Groups 13 (2008), 471–493.
  • D. Cox, The functor of a smooth toric variety, Tohoku Math. J. 47 (1995), 251–262.
  • D. Cox and C. von Renesse, Primitive collections and toric varieties, Tohoku Math. J. 61 (2009), 309–332.
  • V. Danilov, The Geometry of toric varieties, Russian Math. Surveys 33 (1978), 97–154.
  • 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.
  • W. Fulton and J. Harris, Representation theory, Springer-Verlag, New York, Grad. Texts in Math. 129, 1991.
  • W. Fulton, Introduction to toric varieties, Ann. of Math. Stud. 131, Princeton University Press, Princeton, NJ, 1993.
  • L. Gerritzen, F. Herrlich and M. van der Put, Stable $n$-pointed trees of projective lines, Nederl. Akad. Wetensch., Indag. Math. 50 (1988), 131–163.
  • B. Hassett, Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), 316–352.
  • M. Kapranov, Chow quotients of Grassmannians I, Adv. Soviet Math. 16 (1993), 29–110.
  • A. Klyachko, Orbits of a maximal torus on a flag space, Functional Anal. Appl. 19 (1985), 65–66.
  • A. Klyachko, Toric varieties and flag spaces, Trudy Mat. Inst. Steklov. 208 (1995), Teor. Chisel, Algebra i Algebr. Geom., 139–162.
  • F. Knudsen, The projectivity of the moduli space of stable curves II: The stacks $M_{g,n}$, Math. Scand. 52 (1983), 161–199.
  • A. Losev and Yu. Manin, New moduli spaces of pointed curves and pencils of flat connections, Michigan Math. J. 48 (2000), 443–472.
  • C. Procesi, The toric variety associated to Weyl chambers, Mots, 153–161, Lang. Raison. Calc., Hermès, Paris, 1990.
  • S. Shadrin and D. Zvonkine, A group action on Losev-Manin cohomological field theories, arXiv:0909.0800.
  • J. Stembridge, Some permutation representations of Weyl groups associated with the cohomology of toric varieties, Adv. Math. 106 (1994), 244–301.