Conformal blocks and rational normal curves
Author:
Noah Giansiracusa
Journal:
J. Algebraic Geom. 22 (2013), 773-793
DOI:
https://doi.org/10.1090/S1056-3911-2013-00601-3
Published electronically:
June 18, 2013
MathSciNet review:
3084722
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Abstract |
References |
Additional Information
Abstract: We prove that the Chow quotient parameterizing configurations of $n$ points in $\mathbb {P}^d$ which generically lie on a rational normal curve is isomorphic to $\overline {\mathcal {M}}_{0,n}$, generalizing the well-known $d=1$ result of Kapranov. In particular, $\overline {\mathcal {M}}_{0,n}$ admits birational morphisms to all the corresponding geometric invariant theory (GIT) quotients. For symmetric linearizations, the polarization on each GIT quotient pulls back to a divisor that spans the same extremal ray in the symmetric nef cone of $\overline {\mathcal {M}}_{0,n}$ as a conformal blocks line bundle. A symmetry in conformal blocks implies a duality of point-configurations that comes from Gale duality and generalizes a result of Goppa in algebraic coding theory. In a suitable sense, $\overline {\mathcal {M}}_{0,2m}$ is fixed pointwise by the Gale transform when $d=m-1$ so stable curves correspond to self-associated configurations.
References
- Valery Alexeev and David Swinarski, Nef divisors on $\overline M_{0,n}$ from GIT, Geometry and arithmetic, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2012, pp. 1–21 (English, with English and Russian summaries). MR 2987650, DOI https://doi.org/10.4171/119-1/1
- Maxim Arap, Angela Gibney, James Stankewicz, and David Swinarski, $sl_n$ level 1 conformal blocks divisors on $\overline M_{0,n}$, Int. Math. Res. Not. IMRN 7 (2012), 1634–1680. MR 2913186, DOI https://doi.org/10.1093/imrn/rnr064
- D. Avritzer and H. Lange, The moduli spaces of hyperelliptic curves and binary forms, Math. Z. 242 (2002), no. 4, 615–632. MR 1981190, DOI https://doi.org/10.1007/s002090100370
- Arnaud Beauville, Conformal blocks, fusion rules and the Verlinde formula, 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. MR 1360497
- Arnaud Beauville and Yves Laszlo, Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), no. 2, 385–419. MR 1289330
- Michele Bolognesi, Forgetful linear systems on the projective space and rational normal curves over $\scr M^{\rm GIT}_{0,2n}$, Bull. Lond. Math. Soc. 43 (2011), no. 3, 583–596. MR 2820147, DOI https://doi.org/10.1112/blms/bdq125
- P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 5–89. MR 849651
- Igor V. Dolgachev and Yi Hu, Variation of geometric invariant theory quotients, Inst. Hautes Études Sci. Publ. Math. 87 (1998), 5–56. With an appendix by Nicolas Ressayre. MR 1659282
- Igor Dolgachev and David Ortland, Point sets in projective spaces and theta functions, Astérisque 165 (1988), 210 pp. (1989) (English, with French summary). MR 1007155
- David Eisenbud and Sorin Popescu, The projective geometry of the Gale transform, J. Algebra 230 (2000), no. 1, 127–173. MR 1774761, DOI https://doi.org/10.1006/jabr.1999.7940
- Najmuddin Fakhruddin, Chern classes of conformal blocks, Compact moduli spaces and vector bundles, Contemp. Math., vol. 564, Amer. Math. Soc., Providence, RI, 2012, pp. 145–176. MR 2894632, DOI https://doi.org/10.1090/conm/564/11148
- Flaminio Flamini, Towards an inductive construction of self-associated sets of points, Matematiche (Catania) 53 (1998), no. suppl., 33–41 (1999). Pragmatic 1997 (Catania). MR 1696016
- I. M. Gel′fand and R. D. MacPherson, Geometry in Grassmannians and a generalization of the dilogarithm, Adv. in Math. 44 (1982), no. 3, 279–312. MR 658730, DOI https://doi.org/10.1016/0001-8708%2882%2990040-8
- L. Gerritzen, F. Herrlich, and M. van der Put, Stable $n$-pointed trees of projective lines, Nederl. Akad. Wetensch. Indag. Math. 50 (1988), no. 2, 131–163. MR 952512
- Giansiracusa, N. and Gillam, W.D. “On Kapranov’s description of $\overline {\mathcal {M}}_{0,n}$ as a Chow quotient.” math.AG/1103.4661
- Giansiracusa, N. and Simpson, M. “Conic Compactifications of $\overline {\mathcal {M}}_{0,n}$.” Int. Math. Res. Notices Vol. 2010, doi:10.1093/imrn/rnq228 (2010).
- Grothendieck, A. and Dieudonné, J. Éléments de géométrie algébrique. Pub. Math. IHES, 1960.
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- Brendan Hassett, Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), no. 2, 316–352. MR 1957831, DOI https://doi.org/10.1016/S0001-8708%2802%2900058-0
- Benjamin Howard, John Millson, Andrew Snowden, and Ravi Vakil, The ideal of relations for the ring of invariants of $n$ points on the line, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 1, 1–60. MR 2862033, DOI https://doi.org/10.4171/JEMS/295
- Yi Hu, Moduli spaces of stable polygons and symplectic structures on $\overline {\scr M}{}_{0,n}$, Compositio Math. 118 (1999), no. 2, 159–187. MR 1713309, DOI https://doi.org/10.1023/A%3A1001055409867
- Yi Hu, Topological aspects of Chow quotients, J. Differential Geom. 69 (2005), no. 3, 399–440. MR 2170276
- Yi Hu, Stable configurations of linear subspaces and quotient coherent sheaves, Q. J. Pure Appl. Math. 1 (2005), no. 1, 127–164. MR 2154335
- Michael Kapovich and John J. Millson, The symplectic geometry of polygons in Euclidean space, J. Differential Geom. 44 (1996), no. 3, 479–513. MR 1431002
- Keel, S. and J. McKernan. “Contractible extremal rays on $\overline {\mathcal {M}}_{0,n}$.” math.AG/ 9607009.
- M. M. Kapranov, Chow quotients of Grassmannians. I, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 29–110. MR 1237834
- János Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. MR 1440180
- Looijenga, E. “Conformal blocks revisited.” math.AG/0507086.
- D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906
- Marzia Polito, ${\rm SL}(2,\mathbf C)$-quotients de $(\mathbf P^1)^n$, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 12, 1577–1582 (French, with English and French summaries). MR 1367810
- Matthew Simpson, On log canonical models of the moduli space of stable pointed genus zero curves, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–Rice University. MR 2711642
- Akihiro Tsuchiya, Kenji Ueno, and Yasuhiko Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, Integrable systems in quantum field theory and statistical mechanics, Adv. Stud. Pure Math., vol. 19, Academic Press, Boston, MA, 1989, pp. 459–566. MR 1048605, DOI https://doi.org/10.2969/aspm/01910459
References
- Alexeev, V. and Swinarski, D. “Nef divisors on $\overline {\mathcal {M}}_{0,n}$.” Geometry and arithmetic, 1–21, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2012. MR 2987650
- Arap, M., Gibney, A., Stankewicz, J., and Swinarski, D. “$\mathfrak {sl}_n$ level 1 conformal blocks divisors on $\overline {\mathcal {M}}_{0,n}$.” Int. Math. Res. Not. IMRN, 2012, no. 7, 1634–1680. MR 2913186
- Avritzer, D. and Lange, H. “The moduli space of hyperelliptic curves and binary forms.” Math. Z. 242 no. 4 (2002), 615–632. MR 1981190 (2004c:14051)
- Beauville, A. “Conformal Blocks, Fusion Rules, and the Verlinde formula.” Israel Math. Conf. Proc. Bar-Ilan Univ., Ramat Gan, 9 (1996), 75–96. MR 1360497 (97f:17025)
- Beauville, A., and Laszlo, Y. “Conformal Blocks and Generalized Theta Functions.” Commun. Math. Phys. 164 (1994), 385–419. MR 1289330 (95k:14011)
- Bolognesi, M. “Forgetful linear systems on the projective space and rational normal curves over $M_{0,2n}^{GIT}$”. Bull. Lond. Math. Soc. 43 (2011), no. 3, 583–596. MR 2820147 (2012m:14049)
- Deligne, P. and Mostow, G. “Monodromy of hypergeometric functions and non-lattice integral monodromy.” Publications of IHES 63 (1986), 5–90. MR 849651 (88a:22023a)
- Dolgachev, I. and Hu, Y. “Variation of geometric invariant theory quotients.” Inst. Hautes Études Sci. Publ. Math., no. 87 (1998), 5-56. MR 1659282 (2000b:14060)
- Dolgachev, I. and Ortland, D. “Points sets in projective spaces and theta functions.” Asterisque 165 (1988), 1–210. MR 1007155 (90i:14009)
- Eisenbud, D. and Popescu, S. “The projective geometry of the Gale transform.” J. Algebra 230 no. 1 (2000), 127–173. MR 1774761 (2001g:14083)
- Fakhruddin, N. “Chern Classes of Conformal Blocks.” Compact moduli spaces and vector bundles, 145–176, Contemp. Math., 564, Amer. Math. Soc., Providence, RI, 2012. MR 2894632
- Flamini, F. “Towards an inductive construction of self-associated sets of points.” Le Matematiche LIII (1998), 33–41. MR 1696016 (2000m:14055)
- Gelfand, I. and R. MacPherson. “Geometry in Grassmannians and a generalization of the dilogarithm.” Adv. in Math. 44 (1982), 279–312. MR 658730 (84b:57014)
- Gerritzen, L., Herrlich, F., and van der Put, M. “Stable $n$-pointed trees of projective lines.” Indag. Math. 50 (1988), 131–163. MR 952512 (89i:14005)
- Giansiracusa, N. and Gillam, W.D. “On Kapranov’s description of $\overline {\mathcal {M}}_{0,n}$ as a Chow quotient.” math.AG/1103.4661
- Giansiracusa, N. and Simpson, M. “Conic Compactifications of $\overline {\mathcal {M}}_{0,n}$.” Int. Math. Res. Notices Vol. 2010, doi:10.1093/imrn/rnq228 (2010).
- Grothendieck, A. and Dieudonné, J. Éléments de géométrie algébrique. Pub. Math. IHES, 1960.
- Hartshorne, R. Algebraic Geometry. Springer-Verlag GTM 52, 1977. MR 0463157 (57:3116)
- Hassett, B. “Moduli spaces of weighted pointed stable curves.” Adv. Math. 173 no. 2 (2003), 316–352. MR 1957831 (2004b:14040)
- Howard, B., Millson, J., Snowden, A. and Vakil, R. “The ideal of relations for the ring of invariants of n points on the line.” J. Eur. Math. Soc. 14 (2012), no. 1, 1–60. MR 2862033 (2012m:13007)
- Hu, Y. “Moduli spaces of stable polygons and symplectic structures on $\overline {\mathcal {M}}_{0,n}$.” Compos. Math. 118 (1999), 159–187. MR 1713309 (2000g:14018)
- Hu, Y. “Topological Aspects of Chow Quotients.” J. Differential Geometry 69 (2005), 399–440. MR 2170276 (2006f:14054)
- Hu, Y. “Stable configurations of linear subspaces and quotient coherent sheaves.” Q. J. Pure Appl. Math. 1 (2005), 127–164. MR 2154335 (2007c:14048)
- Kapovich, M. and J. Millson. “The symplectic geometry of polygons in Euclidean space.” J. Differential Geom. 44 no. 3 (1996), 479–513. MR 1431002 (98a:58027)
- Keel, S. and J. McKernan. “Contractible extremal rays on $\overline {\mathcal {M}}_{0,n}$.” math.AG/ 9607009.
- Kapranov, M. “Chow quotients of Grassmannians, I.” Adv. Sov. Math. 16 no. 2 (1993), 29–110. MR 1237834 (95g:14053)
- Kollár, J. Rational curves on algebraic varieties. Springer, Secaucus, NJ, 1996. MR 1440180 (98c:14001)
- Looijenga, E. “Conformal blocks revisited.” math.AG/0507086.
- Mumford, D., Fogarty, J. and F. Kirwan. Geometric Invariant Theory. Third Edition. Springer, 1994. MR 1304906 (95m:14012)
- Polito, M. “$\text {SL}(2,\mathbb {C})$-quotients de $(\mathbb {P}^1)^n$.” C.R. Acad. Sci Paris 321 Série I (1995), 1577-1582. MR 1367810 (97a:14049)
- Simpson, M. “On Log Canonical Models of the Moduli Space of Stable Pointed Genus Zero Curves.” Ph.D dissertation, Rice University, 2008. MR 2711642
- Tsuchiya, A., Ueno, K., and Y. Yamada. “Conformal field theory on universal family of stable curves with gauge symmetries.” Adv. Stud. Pure Math. 19 (1989), 459–566. MR 1048605 (92a:81191)
Additional Information
Noah Giansiracusa
Affiliation:
Department of Mathematics, Brown University, Providence, Rhode Island 02912
Address at time of publication:
Department of Mathematics, University of California, Berkeley, Berkeley, California 94720
Email:
noahgian@math.brown.edu, noahgian@math.berkeley.edu
Received by editor(s):
January 16, 2011
Received by editor(s) in revised form:
April 11, 2011
Published electronically:
June 18, 2013
Additional Notes:
The author was partially supported by funds from NSF DMS-0901278.
Communicated by:
Michel Brion
Article copyright:
© Copyright 2013
University Press, Inc.
The copyright for this article reverts to public domain 28 years after publication.