Skip to main content Accessibility help
×
Home
Hostname: page-component-78bd46657c-2z7pd Total loading time: 0.238 Render date: 2021-05-08T11:59:43.567Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Higher-Level Conformal Blocks Divisors on

Published online by Cambridge University Press:  16 January 2014

Valery Alexeev
Affiliation:
Department of Mathematics, The University of Georgia, Athens, GA 30602, USA (valery@math.uga.edu; agibney@math.uga.edu)
Angela Gibney
Affiliation:
Department of Mathematics, The University of Georgia, Athens, GA 30602, USA (valery@math.uga.edu; agibney@math.uga.edu)
David Swinarski
Affiliation:
Department of Mathematics, Lincoln Center Campus, Fordham University, New York, NY 10023, USA (dswinarski@fordham.edu)

Abstract

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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2014 

Access options

Get access to the full version of this content by using one of the access options below.

References

1.Arakelov, S. Ju., Families of algebraic curves with fixed degeneracies, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 12691293.Google Scholar
2.Arap, M., Gibney, A., Stankewicz, J. and Swinarski, D., level conformai blocks divisors on , Int. Math. Res. Not. 2012(7) (2012), 16341680.Google Scholar
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).Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
6.Dolgachev, I. V. and Hu, Y., Variation of geometric invariant theory quotients, Publ. Math. IHES 87(1) (1998), 556.CrossRefGoogle Scholar
7.Faber, C., Chow rings of moduli spaces of curves, I, The Chow ring of , Annals Math. 132(2) (1990), 331419.CrossRefGoogle Scholar
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).CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
10.Fedorchuk, M., Cyclic covering morphisms on , eprint (arXiv:1105.0655, 2011).Google Scholar
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).CrossRefGoogle Scholar
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 www.math.tu-berlin.de/polymake).CrossRefGoogle Scholar
13.Giansiracusa, N., Conformal blocks and rational normal curves, J. Alg. Geom. 22 (2013), 773793.CrossRefGoogle Scholar
14.Giansiracusa, N. and Gibney, A., The cone of type A, level 1, conformal blocks divisors, Adv. Math. 231(2) (2012), 798814.CrossRefGoogle Scholar
15.Giansiracusa, N. and Simpson, M., GIT compactifications of M0,n from conics, Int. Math. Res. Not. 2011 (14) (2011), 33153334.Google Scholar
16.Gibney, A., Numerical criteria for divisors on to be ample, Compositio Math. 145(5) (2009), 12271248.CrossRefGoogle Scholar
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).CrossRefGoogle Scholar
18.Gibney, A. and Krashen, D., NEFWIZ: software for divisors on the moduli space of curves, Version 1.1 (2006).Google Scholar
19.Gibney, A., Keel, S. and Morrison, I., Towards the ample cone of , J. Am. Math. Soc. 15(2) (2002), 273294.CrossRefGoogle Scholar
20.Grayson, D. and Stillman, M., Macaulay2: a software system for research in algebraic geometry, Version 1.1 (2008) (available at www.math.uiuc.edu/Macaulay2).Google Scholar
21.Hassett, B., Moduli spaces of weighted pointed stable curves, Adv. Math. 173(2) (2003), 316352.CrossRefGoogle Scholar
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).CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
24.Hyeon, D. and Lee, Y., Stability of bicanonical curves of genus three, J. Pure Appl. Alg. 213(10) (2009), 19912000.CrossRefGoogle Scholar
25.Keel, S., Basepoint freeness for nef and big line bundles in positive characteristic, Annals Math. 149(1) (1999), 253286.CrossRefGoogle Scholar
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).Google Scholar
27.Looijenga, E., Conformal blocks revisited, eprint (arXiv:math/0507086v1, 2005).Google Scholar
28.Pandharipande, R., The canonical class of (ℙr, d) and enumerative geometry, Int. Math. Res. Not. 1997(4) (1997), 173186.Google Scholar
29.Rulla, W., The birational geometry of and , PhD thesis, University of Texas (2001).Google Scholar
30.Schubert, D., A new compactification of the moduli space of curves, Compositio Math. 78(3) (1991), 297313.Google Scholar
31.Swinarski, D., Conformal Blocks: software for computing conformal block divisors in Macaulay2, Version 1.1 (2011) (available at www.math.uga.edu/~davids/conformalblocks).Google Scholar
32.Thaddeus, M., Geometric invariant theory and flips, J. Am. Math. Soc. 9(3) (1996), 691723.CrossRefGoogle Scholar
33.Ueno, K., Conformal field theory with gauge symmetry, Fields Institute Monographs, Volume 24 (American Mathematical Society, Providence, RI, 2008).CrossRefGoogle Scholar

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Higher-Level Conformal Blocks Divisors on
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Higher-Level Conformal Blocks Divisors on
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Higher-Level Conformal Blocks Divisors on
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *