NONVANISHING OF CONFORMAL BLOCKS DIVISORS ON \( {\overline{\mathrm{M}}}_{0,n} \)


We introduce and study the problem of finding necessary and sufficient conditions under which a conformal blocks divisor on \( {\overline{\mathrm{M}}}_{0,n} \) is nonzero, solving the problem completely for \( \mathfrak{s}{\mathfrak{l}}_2 \). We give necessary nonvanishing conditions in type A, which are sufficient when theta and critical levels coincide. We also show divisors are subject to additive identities, reflecting a decomposition of the weights and level.

This is a preview of subscription content, access via your institution.


  1. [AGS14]

    V. Alexeev, A. Gibney, D. Swinarski, Higher-level \( \mathfrak{s}{\mathfrak{l}}_2 \) conformal blocks divisors on \( {\overline{\mathrm{M}}}_{0,n} \), Proc. Edinb. Math. Soc. (2) 57 (2014), no. 1, 7–30.

  2. [Bea96]

    A. Beauville, Conformal blocks, fusion rules and the Verlinde formula, in: Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc., Vol. 9, Bar-Ilan Univ., Ramat Gan, 1996, pp. 75–96.

  3. [Bel04]

    P. Belkale, Invariant theory of GL(n) and intersection theory of Grassmannians, Int. Math. Res. Not. 69 (2004), 3709–3721.

  4. [Bel07]

    P. Belkale, Geometric proof of a conjecture of Fulton, Adv. Math. 216 (2007), no. 1, 346–357.

  5. [Bel08]

    P. Belkale, Quantum generalization of the Horn conjecture, J. Amer. Math. Soc. 21 (2008), no. 2, 365–408.

  6. [BGM15]

    P. Belkale, A. Gibney, S. Mukhopadhyay, Vanishing and identities of conformal blocks divisors, Algebr. Geom. 2 (2015), no. 1, 62–90.

  7. [BK13]

    P. Belkale, S. Kumar, The multiplicative eigenvalue problem and deformed quantum cohomology, Adv. Math. (2015), DOI: 10.1016/j.aim.2015.09.034, arXiv:1310.3191 (2013).

  8. [CT15]

    A.-M. Castravet, J. Tevelev, \( {\overline{\mathrm{M}}}_{0,n} \) is not a Mori dream space, Duke Math. J. 164 (2015), no 8, 1641–1667.

  9. [Fak12]

    N. Fakhruddin, Chern classes of conformal blocks, in: Compact Moduli Spaces and Vector Bundles, Contemp. Math., Vol. 564, Amer. Math. Soc., Providence, RI, 2012, pp. 145–176.

  10. [FSV95]

    B. L. Feigin V. V. Schechtman A. N. Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models. II, Comm. Math. Phys. 170 (1995), no. 1, 219–247.

  11. [FH91]

    W. Fulton, J. Harris, Representation Theory, Graduate Texts in Mathematics, Vol. 129, Springer-Verlag, New York, 1991.

  12. [Ful00]

    W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 209–249.

  13. [GG12]

    N. Giansiracusa, A. Gibney, The cone of type A, level 1 conformal block divisors, Adv. Math. 231 (2012), 798–814.

  14. [GJMS13]

    A. Gibney, D. Jensen, H.-B. Moon, D. Swinarski, Veronese quotient models of \( {\overline{\mathrm{M}}}_{0,n} \) and conformal blocks, Michigan Math. J. 62 (2013), no. 4, 721–751.

  15. [Gib09]

    A. Gibney, Numerical criteria for divisors on \( {\overline{\mathrm{M}}}_g \) to be ample, Compos. Math. 145 (2009), no. 5, 1227–1248.

  16. [GK14]

    J. González, K. Karu, Some nonfinitely generated Cox rings, (2014), arXiv: 1407.6344, to appear in Compositio Math.

  17. [Has03]

    B. Hassett, Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), no. 2, 316–352.

  18. [Hob15]

    N. Hobson, Quantum Kostka and the rank one problem for \( \mathfrak{s}{\mathfrak{l}}_{2m} \) (2015), arXiv:1508.06952.

  19. [Kaz14]

    A. Kazanova, On Sn invariant conformal blocks vector bundles of rank one on \( {\overline{\mathrm{M}}}_{0,n} \) (2014), arXiv:1404.5845v1, to appear in Manuscripta Math.

  20. [KM13]

    S. Keel, J. McKernan, James, Contractible extremal rays on \( {\overline{\mathrm{M}}}_{0,n} \), in: Handbook of Moduli, Vol. II, Adv. Lect. Math., Vol. 25, Int. Press, Somerville, MA, 2013, pp. 115–130.

  21. [KTW04]

    A. Knutson, T. Tao, C. Woodward, The honeycomb model of GL n (ℂ) tensor products. II. Puzzles determine facets of the Littlewood–Richardson cone J. Amer. Math. Soc., 17 (2004), no. 1, 19–48.

  22. [Man09]

    C. Manon, The algebra of conformal blocks (2009), arXiv:0910.0577v6.

  23. [Muk14]

    S. Mukhopadhyay, Remarks on level one conformal block divisors, C. R. Math. Acad. Sci. Paris 352 (2014), no. 3, 179–182.

  24. [Laz04]

    R. Lazarsfeld, Positivity in Algebraic Geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A, Vol. 48, Springer-Verlag, Berlin, 2004.

  25. [Oud11]

    R. Oudompheng, Rank-level duality for conformal blocks of the linear group, J. Algebr. Geom. 20 (2011), no. 3, 559–597.

  26. [Pau96]

    C. Pauly, Espaces de modules de fibrés paraboliques et blocs conformes, Duke Math. J. 84 (1996), no. 1, 217–235.

  27. [Sor96]

    C. Sorger, La formule de Verlinde, in: Séminaire Bourbaki, Vol. 1994/95, Astérisque 237 (1996), Exp. no. 794, 3, pp. 87–114.

  28. [Swi10]

    D. Swinarski, ConformalBlocks: a Macaulay2 package for computing conformal block divisors, (2010),

  29. [Swi11]

    D. Swinarski, \( \mathfrak{s}{\mathfrak{l}}_2 \) conformal block divisors and the nef cone of \( {\overline{\mathrm{M}}}_{0,n} \) (2011), arXiv:1107.5331.

  30. [Tsu93]

    Y. Tsuchimoto, On the coordinate-free description of conformal blocks, J. Math. Kyoto. Univ. 1 (1993), 29–49.

  31. [TUY89]

    A. Tsuchiya, K. Ueno, Y. Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, in: Integrable Systems in Quantum Field Theory and Statistical Mechanics, Adv. Stud. Pure Math., Vol. 19, Academic Press, Boston, MA, 1989, pp. 459–566.

  32. [Wit95]

    E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, Geometry, Topology IV (1995), 357–422.

Download references

Author information



Corresponding author

Correspondence to P. BELKALE.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

BELKALE, P., GIBNEY, A. & MUKHOPADHYAY, S. NONVANISHING OF CONFORMAL BLOCKS DIVISORS ON \( {\overline{\mathrm{M}}}_{0,n} \) . Transformation Groups 21, 329–353 (2016).

Download citation


  • Vector Bundle
  • Line Bundle
  • Young Diagram
  • Conformal Block
  • Bundle Versus