Hostname: page-component-54dcc4c588-5q6g5 Total loading time: 0 Render date: 2025-10-05T08:27:09.994Z Has data issue: false hasContentIssue false
  • English
  • Français

Cohomological invariants of root stacks and admissible double coverings

Part of: Cycles and subschemes Curves (Co)homology theory

Published online by Cambridge University Press:  02 November 2021

Andrea Di Lorenzo  and
Roberto Pirisi Open the ORCID record for Roberto Pirisi [Opens in a new window]
Andrea Di Lorenzo
Affiliation:
Department of Mathematics, Humboldt-Universität zu Berlin, Berlin, Germany e-mail: dilorena@hu-berlin.de
Roberto Pirisi*
Affiliation:
Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma, Rome, Italy
*
e-mail: roberto.pirisi@uniroma1.it
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a formula for the cohomological invariants of a root stack, which we apply to compute the cohomological invariants and the Brauer group of the compactification of the stacks of hyperelliptic curves given by admissible double coverings.

Keywords

Cohomological invariantsroot stackshyperelliptic curvesadmissible coverings

MSC classification

Primary: 14F22: Brauer groups of schemes
Secondary: 14H10: Families, moduli (algebraic) 14C15: (Equivariant) Chow groups and rings; motives

Information

Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 1 , February 2023 , pp. 202 - 224
Check for updates
Copyright
© Canadian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abramovich, D., Corti, A., and Vistoli, A., Twisted bundles and admissible covers . Comm. Algebra 31(2003), no. 8, 3547–3618. Special issue in honor of Steven L. Kleiman.Google Scholar
Arbarello, E., Cornalba, M., and Griffiths, P. A., Geometry of algebraic curves. Vol. II, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), 268, Springer, Heidelberg, 2011.Google Scholar
Arsie, A. and Vistoli, A., Stacks of cyclic covers of projective spaces . Compos. Math. 140(2004), no. 3, 647–666.CrossRefGoogle Scholar
Beauville, A., Prym varieties and the Schottky problem . Invent. Math. 41(1977), no. 2.CrossRefGoogle Scholar
Cadman, C., Using stacks to impose tangency conditions on curves . Amer. J. Math. 129(2007), no. 2, 149–196.CrossRefGoogle Scholar
Cornalba, M., The Picard group of the moduli stack of stable hyperelliptic curves . Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 18(2007), no. 1, 109–115.CrossRefGoogle Scholar
Cornalba, M. and Harris, J., Divisor classes associated to families of stable varieties, with applications to the moduli space of curves . Ann. Sci. École Norm. Supér. (4) 21(1988), no. 3, 455–475.CrossRefGoogle Scholar
Di Lorenzo, A. and Pirisi, R., A complete description of the cohomological invariants of even genus hyperelliptic curves . Doc. Math. 26(2021), 199–230.CrossRefGoogle Scholar
Di Lorenzo, A. and Pirisi, R., Brauer groups of moduli of hyperelliptic curves via cohomological invariants. Forum Math. Sigma 9(2021).CrossRefGoogle Scholar
Di Lorenzo, A. and Vistoli, A., Polarized twisted conics and moduli of stable curves of genus two. Preprint, 2021. arXiv:2103.13204 Google Scholar
Edidin, D. and Graham, W., Equivariant intersection theory . Invent. Math. 131(1998), no. 3.CrossRefGoogle Scholar
Garibaldi, S., Merkurjev, A., and Serre, J.-P., Cohomological invariants in Galois cohomology, University Lecture Series, 28, American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
Gille, S. and Hirsch, C., On the splitting principle for cohomological invariants of reflection groups, Trans. Groups, http://doi.org/10.1007/S00031-020-09637-6, arXiv:1908.08146 Google Scholar
Gorchinskiy, S. and Viviani, F., Picard group of moduli of hyperelliptic curves . Math. Z. 258(2008), 319–331.CrossRefGoogle Scholar
Guillot, P., Geometric methods for cohomological invariants . Doc. Math. 12(2007), 521–545.CrossRefGoogle Scholar
Harper, A., Factorization for stacks and boundary complexes. Preprint, 2017. arXiv:1706.07999.Google Scholar
Harris, J. and Mumford, D., On the Kodaira dimension of the moduli space of curves . Invent. Math. 67(1982), 23–86. With an appendix by William Fulton.CrossRefGoogle Scholar
Kausz, I., A discriminant and an upper bound for ${\omega}^2$ for hyperelliptic arithmetic surfaces. Compos. Math. 115(1999) no. 1, 37–69.CrossRefGoogle Scholar
Kresch, A., On the geometry of Deligne–Mumford stacks . Proc. Sympos. Pure Math. 80(2009), 259–271.CrossRefGoogle Scholar
Olsson, M., Algebraic spaces and stacks, Colloquium Publications, 62, American Mathematical Society, Providence, RI, 2016.Google Scholar
Pirisi, R., Cohomological invariants of algebraic stacks . Trans. Amer. Math. Soc. 370(2018), no. 3.Google Scholar
Rost, M., Chow groups with coefficients . Doc. Math. 1(1996), no. 16, 319–393.CrossRefGoogle Scholar
Scavia, F., The stack of admissible double covers, M.Sc. thesis, Scuola Normale Superiore, 2017. https://etd.adm.unipi.it/t/etd-04262017-101615/ Google Scholar
Voevodsky, V., On motivic cohomology with $\mathbb{Z}/ \ell$ -coefficients . Ann. Math. 174(2011), 401–438.CrossRefGoogle Scholar
Witt, E., Theorie der quadratischen Formen in beliebigen Körpern . J. Reine Angew. Math. 176(1937), 31–44 (in German).Google Scholar
Yamaki, K., Cornalba–Harris equality for semistable hyperelliptic curves in positive characteristic . Asian J. Math. 8(2004), no. 3, 409–426.CrossRefGoogle Scholar