Abstract
We study Deligne products for forgetful maps between moduli spaces of marked curves by offering a closed formula for tautological line bundles associated to marked points. In particular, we show that the Deligne products for line bundles on the total spaces corresponding to “forgotten” marked points are positive integral multiples of the Weil-Petersson bundles on the base moduli spaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arbarello E., Cornalba M.: The Picard groups of the moduli spaces of curves. Topology 26(2), 153–171 (1987)
Deligne, P.: Le déterminant de la cohomologie. In: Current trends in arithmetic algebraic geometry, Contemporary Math. Vol. 67, Providence, RI: Amer. Math. Soc., pp. 93–178, 1987
Deligne P., Mumford D.: The irreducibility of the space of curves of given genus. IHES Publ. Math. 36, 75–109 (1969)
Harer J.: The second homology group of the mapping class group of an orientable surface. Invent. Math. 72(2), 221–239 (1983)
Kaufmann R., Manin Yu., Zagier D.: Higher Weil-Petersson volumes of moduli spaces of stable n-pointed curves. Commun. Math. Phys. 181(3), 763–787 (1996)
Knudsen F.: The projectivity of the moduli space of stable curves, II. Math. Scand. 52, 161–199 (1983)
Knudsen F., Mumford D.: The projectivity of the moduli space of stable curves. I. Math. Scand. 39, 19–55 (1976)
Lando, S.K., Zvonkin, A.K.: Graphs on Surfaces and Their Applications, In: Encyclopedia of Mathematical Sciences, Berlin–Heidelberg: Springer-Verlag, 2004
Takhtajan L., Zograf P.: The Selberg zeta function and a new Kähler metric on the moduli space of punctured Riemann surfaces. J. Geo. Phys. 5, 551–570 (1988)
Takhtajan L., Zograf P.: A local index theorem for families of \({\bar\partial}\) -operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaces. Commun. Math. Phys. 137, 399–426 (1991)
Weng, L.: Hyperbolic Metrics, Selberg Zeta Functions and Arakelov Theory for Punctured Riemann Surfaces. Lecture Note Series in Mathematics 6, Osaka: Osaka University, 1998
Weng L.: Ω-admissible theory, II.. Math. Ann. 320, 239–283 (2001)
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Takhtajan
Partially supported by the Japan Society for the Promotion of Science.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Weng, L., Zagier, D. Deligne Products of Line Bundles over Moduli Spaces of Curves. Commun. Math. Phys. 281, 793–803 (2008). https://doi.org/10.1007/s00220-008-0494-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-008-0494-5