Abstract
We show the compactly supported motive of the moduli stack of degree n rational curves on the weighted projective stack \({\mathcal {P}}(a,b)\) is of mixed Tate type over any base field K with \(\hbox {char}(K) \not \mid a,b\) and has class \({\mathbb {L}}^{(a+b)n+1}-{\mathbb {L}}^{(a+b)n-1}\) in the Grothendieck ring of stacks. In particular, this improves upon the results of (Han and Park in Math Ann 375(3–4), 1745–1760, 2019) regarding the arithmetic invariant of the moduli stack \({\mathcal {L}}_{1,12n} :=\mathrm {Hom}_{n}({\mathbb {P}}^1, \overline{{\mathcal {M}}}_{1,1})\) of stable elliptic fibrations over \({\mathbb {P}}^{1}\) with 12n nodal singular fibers and a marked Weierstrass section.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramovich, D., Olsson, M., Vistoli, A.: Tame stacks in positive characteristic. Annales de l’Institut Fourier 58(4), 1057–1091 (2008)
Cisinski, D.-C., Déglise, F.: Integral Mixed Motives in Equal Characteristic. Documenta Mathematica, Extra Volume: Alexander S. Merkurjev’s Sixtieth Birthday, pp. 145–194 (2015)
Ekedahl, T.: The Grothendieck group of algebraic stacks, arXiv:0903.3143 (2009)
Farb, B., Wolfson, J.: Topology and arithmetic of resultants, I. N. Y. J. Math. 22, 801–821 (2016)
Farb, B., Wolfson, J.: Corrigendum to "Topology and arithmetic of resultants, I". N. Y. J. Math. 25, 195–197 (2019)
Hassett, B.: Classical and minimal models of the moduli space of curves of genus two. Geometric Methods in Algebra and Number Theory, Progress in Mathematics, 235, Birkhäuser Boston, pp. 169–192 (2005)
Han, C., Park, J.: Arithmetic of the moduli of semistable elliptic surfaces. Math. Ann. 375(3–4), 1745–1760 (2019)
Horel, G.: Motivic homological stability for configuration spaces of the line. Bull. Lond. Math. Soc. 48(4), 601–616 (2016)
Kelly, S.: Triangulated categories of motives in positive characteristic, Ph.D. thesis, University of Paris 13; Australian National University. arXiv:1305.5349 (2013)
Mazza, C., Voevodsky, V., Weibel, C.: Lecture Notes on Motivic Cohomology. Clay Mathematics Monographs, vol. 2. American Mathematical Society, Providence (2006)
Olsson, M.: Hom-stacks and restriction of scalars. Duke Math. J. 134(1), 139–164 (2006)
Park, J.: \(\ell \)-adic étale cohomology of the moduli of elliptic & hyperelliptic fibrations, arXiv:1812.11694 (2018)
Segal, G.: The topology of spaces of rational functions. Acta Math. 143, 39–72 (1979)
Totaro, B.: The motive of a classifying space. Geom. Topol. 20(4), 2079–2133 (2016)
Totaro, B.: Chow groups, Chow cohomology, and linear varieties, Forum of Mathematics, Sigma, 2, Paper No. E17, p 25 (2014)
Voevodsky, V.: Triangulated Categories of Motives over a Field, “Cycles, Transfers, and Motivic Homology Theories”. Annals of Mathematics Studies, vol. 143, pp. 188–238. Princeton University Press, Princeton, NJ (2000)
Acknowledgements
We would like to thank the organizers of the Pacific Institute for the Mathematical Sciences Workshop on Arithmetic Topology at University of British Columbia in June 2019 for their excellent hospitality and inspiring conference where most of this work was done. Special thanks to Dori Bejleri, Benson Farb, Changho Han, Craig Westerland and Jesse Wolfson for many helpful conversations. Jun-Yong Park was supported by IBS-R003-D1, Institute for Basic Science in Korea. Finally, we thank the reviewer for their in-depth and helpful suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Park, J., Spink, H. Motive of the moduli stack of rational curves on a weighted projective stack. Res Math Sci 8, 1 (2021). https://doi.org/10.1007/s40687-020-00236-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40687-020-00236-1