Abstract
The main subject of this work is the difference between the coarse moduli space and the stack of hyperelliptic curves. We compute their Picard groups, giving explicit description of the generators. We get an application to the non-existence of a tautological family over the coarse moduli space.
Similar content being viewed by others
References
Arbarello E. and Cornalba M. (1987). The Picard groups of the moduli space of curves. Topology 26: 153–171
Arsie A. and Vistoli A. (2004). Stacks of cyclic covers of projective spaces. Compos. Math. 140: 647–666
Cornalba, M.: The Picard group of the moduli stack of stable hyperelliptic curves. Preprint available at math.AG/0605531
Edidin D. and Graham W. (1998). Equivariant intersection theory. Invent. Math. 131: 595–634
Gorchinskiy, S., Viviani, F.: Families of hyperelliptic curves. Preprint available at math.AG/0511627
Harer J. (1983). The second homology group of the mapping class group of an orientable surface. Invent. Math. 72: 221–239
Harris J. and Morrison D. (1998). Moduli of Curves GTM 187. Springer, New York
Igusa J. (1960). Arithmetic variety of moduli for genus two. Ann. of Math. 72: 612–649
Lonsted K. (1984). The singular points on the moduli spaces for smooth curves. Math. Ann. 266: 397–402
Lonsted K. and Kleiman S. (1979). Basics on families of hyperelliptic curves. Comp. Math. 38: 83–111
Mestrano N. and Ramanan S. (1985). Poincaré bundles for families of curves. J. Reine Angew. Math. 362: 169–178
Mumford, D.: Picard groups of moduli problems. In: Proceedings and Conference on Arithmetical Algebraic Geometry, Purdue University, Harper and Row, New York 33–81 (1963)
Oort F. (1975). Singularities of the moduli scheme for curves of genus three. Indag. Math. 37: 170–174
Popp H. (1969). The singularities of the moduli schemes of curves. J. Number Theory 1: 90–107
Rauch H.E. (1962). The singularities of the modulus space. Bull. Am. Math. Soc. 68: 390–394
Vistoli A. (1998). The Chow ring of \({\mathcal{M}}_{2}\) Invent. Math. 131: 635–644
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was supported by RFFI grants no. 04-01-00613, 05-01-00455 and INTAS grant no. 05-1000008-8118.
The second author was supported by a grant from Mittag-Leffler Institute.
Rights and permissions
About this article
Cite this article
Gorchinskiy, S., Viviani, F. Picard group of moduli of hyperelliptic curves. Math. Z. 258, 319–331 (2008). https://doi.org/10.1007/s00209-007-0173-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-007-0173-9