# Motive of the moduli stack of rational curves on a weighted projective stack

Motive of the moduli stack of rational curves on a weighted projective stack We show the compactly supported motive of the moduli stack of degree n rational curves on the weighted projective stack P(a,b)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {P}}(a,b)$$\end{document} is of mixed Tate type over any base field K with char(K)∤a,b\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\hbox {char}(K) \not \mid a,b$$\end{document} and has class L(a+b)n+1-L(a+b)n-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {L}}^{(a+b)n+1}-{\mathbb {L}}^{(a+b)n-1}$$\end{document} 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 L1,12n:=Homn(P1,M¯1,1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {L}}_{1,12n} :=\mathrm {Hom}_{n}({\mathbb {P}}^1, \overline{{\mathcal {M}}}_{1,1})$$\end{document} of stable elliptic fibrations over P1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {P}}^{1}$$\end{document} with 12n nodal singular fibers and a marked Weierstrass section. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

# Motive of the moduli stack of rational curves on a weighted projective stack

, Volume 8 (1) – Jan 3, 2021
8 pages

/lp/springer-journals/motive-of-the-moduli-stack-of-rational-curves-on-a-weighted-projective-lQ7XxHniNo
Publisher
Springer Journals
eISSN
2197-9847
DOI
10.1007/s40687-020-00236-1
Publisher site
See Article on Publisher Site

### Abstract

We show the compactly supported motive of the moduli stack of degree n rational curves on the weighted projective stack P(a,b)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {P}}(a,b)$$\end{document} is of mixed Tate type over any base field K with char(K)∤a,b\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\hbox {char}(K) \not \mid a,b$$\end{document} and has class L(a+b)n+1-L(a+b)n-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {L}}^{(a+b)n+1}-{\mathbb {L}}^{(a+b)n-1}$$\end{document} 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 L1,12n:=Homn(P1,M¯1,1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {L}}_{1,12n} :=\mathrm {Hom}_{n}({\mathbb {P}}^1, \overline{{\mathcal {M}}}_{1,1})$$\end{document} of stable elliptic fibrations over P1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {P}}^{1}$$\end{document} with 12n nodal singular fibers and a marked Weierstrass section.

### Journal

Research in the Mathematical SciencesSpringer Journals

Published: Jan 3, 2021