Skip to main content Accessibility help

Higher-Level Conformal Blocks Divisors on

  • Valery Alexeev (a1), Angela Gibney (a1) and David Swinarski (a2)


We study a family of semi-ample divisors on the moduli space of n-pointed genus 0 curves given by higher-level conformal blocks. We derive formulae for their intersections with a basis of 1-cycles, show that they form a basis for the Sn-invariant Picard group, and generate a full-dimensional subcone of the Sn-invariant nef cone. We find their position in the nef cone and study their associated morphisms.



Hide All
1.Arakelov, S. Ju., Families of algebraic curves with fixed degeneracies, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 12691293.
2.Arap, M., Gibney, A., Stankewicz, J. and Swinarski, D., level conformai blocks divisors on , Int. Math. Res. Not. 2012(7) (2012), 16341680.
3.Beauville, A., Conformal blocks, fusion rules and the Verlinde formula, in Proc. Hirzebruch 65 Conf. on Algebraic Geometry, May 2-7, 1993, Israel Mathematical Conference Proceedings, Volume 9, pp. 7596 (Bar-Ilan University, Ramat Gan, 1996).
4.Birkar, C., Cascini, P., Hacon, C. D. and McKernan, J., Existence of minimal models for varieties of log general type, J. Am. Math. Soc. 23 (2010), 405468.
5.Cornalba, M. and Harris, J., Divisor classes associated to families of stable varieties, with applications to the moduli space of curves, Annales Scient. Éc. Norm. Sup. 21(3) (1988), 455475.
6.Dolgachev, I. V. and Hu, Y., Variation of geometric invariant theory quotients, Publ. Math. IHES 87(1) (1998), 556.
7.Faber, C., Chow rings of moduli spaces of curves, I, The Chow ring of , Annals Math. 132(2) (1990), 331419.
8.Fakhruddin, N., Chern classes of conformal blocks, in Compact moduli spaces and vector bundles, Contemporary Mathematics, Volume 564, pp. 145176 (American Mathematical Society, Providence, RI, 2012).
9.Farkas, G. and Gibney, A., The Mori cones of moduli spaces of pointed curves of small genus, Trans. Am. Math. Soc. 355(3) (2003), 11831199.
10.Fedorchuk, M., Cyclic covering morphisms on , eprint (arXiv:1105.0655, 2011).
11.Feingold, A. J., Fusion rules for affine Kac–Moody algebras, in Kac–Moody Lie Algebras and Related Topics: Proc. Ramanujan Int. Symp. on Kac–Moody Lie Algebras and Applications, January 28–31, 2002, Contemporary Mathematics, Volume 343, pp. 5396 (American Mathematical Society, Providence, RI, 2004).
12.Gawrilow, E. and Joswig, M., POLYMAKE: a framework for analyzing convex polytopes, Version 2.3, in Polytopes: combinatorics and computation, Deutsche Mathematiker-Vereinigung Seminar, Volume 29, pp. 4373 (Birkhäuser, 2000) (available at
13.Giansiracusa, N., Conformal blocks and rational normal curves, J. Alg. Geom. 22 (2013), 773793.
14.Giansiracusa, N. and Gibney, A., The cone of type A, level 1, conformal blocks divisors, Adv. Math. 231(2) (2012), 798814.
15.Giansiracusa, N. and Simpson, M., GIT compactifications of M0,n from conics, Int. Math. Res. Not. 2011 (14) (2011), 33153334.
16.Gibney, A., Numerical criteria for divisors on to be ample, Compositio Math. 145(5) (2009), 12271248.
17.Gibney, A., On extensions of the Torelli map, in Geometry and arithmetic, European Mathematical Society Series of Congress Reports, pp. 125136 (European Mathematical Society, Zurich, 2012).
18.Gibney, A. and Krashen, D., NEFWIZ: software for divisors on the moduli space of curves, Version 1.1 (2006).
19.Gibney, A., Keel, S. and Morrison, I., Towards the ample cone of , J. Am. Math. Soc. 15(2) (2002), 273294.
20.Grayson, D. and Stillman, M., Macaulay2: a software system for research in algebraic geometry, Version 1.1 (2008) (available at
21.Hassett, B., Moduli spaces of weighted pointed stable curves, Adv. Math. 173(2) (2003), 316352.
22.Hassett, B., Classical and minimal models of the moduli space of curves of genus two, in Geometric methods in algebra and number theory, Progress in Mathematics, Volume 235, pp. 169192 (Birkhäuser, 2005).
23.Hassett, B. and Hyeon, D., Log canonical models for the moduli space of curves: the first divisorial contraction, Trans. Am. Math. Soc. 361(8) (2009), 44714489.
24.Hyeon, D. and Lee, Y., Stability of bicanonical curves of genus three, J. Pure Appl. Alg. 213(10) (2009), 19912000.
25.Keel, S., Basepoint freeness for nef and big line bundles in positive characteristic, Annals Math. 149(1) (1999), 253286.
26.Keel, S. and Mckernan, J., Contractible extremal rays on , in Handbook of moduli, Volume II (ed. Farkas, G. and Morrison, I.), Advanced Lectures in Mathematics, Volume 25 (International Press, Somerville, MA, 2013).
27.Looijenga, E., Conformal blocks revisited, eprint (arXiv:math/0507086v1, 2005).
28.Pandharipande, R., The canonical class of (ℙr, d) and enumerative geometry, Int. Math. Res. Not. 1997(4) (1997), 173186.
29.Rulla, W., The birational geometry of and , PhD thesis, University of Texas (2001).
30.Schubert, D., A new compactification of the moduli space of curves, Compositio Math. 78(3) (1991), 297313.
31.Swinarski, D., Conformal Blocks: software for computing conformal block divisors in Macaulay2, Version 1.1 (2011) (available at
32.Thaddeus, M., Geometric invariant theory and flips, J. Am. Math. Soc. 9(3) (1996), 691723.
33.Ueno, K., Conformal field theory with gauge symmetry, Fields Institute Monographs, Volume 24 (American Mathematical Society, Providence, RI, 2008).


Higher-Level Conformal Blocks Divisors on

  • Valery Alexeev (a1), Angela Gibney (a1) and David Swinarski (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed