Geodesic
On the Motivic Class of the Moduli Stack of Twisted $G$-Covers
Symmetry, integrability and geometry: methods and applications, Tome 19 (2023) Cet article a éte moissonné depuis la source Math-Net.Ru

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We describe the class, in the Grothendieck group of stacks, of the stack of twisted $G$-covers of genus $0$ curves, in terms of the loci corresponding to covers over smooth bases.
Keywords: moduli spaces of covers, Grothendieck group of stacks. 
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     author = {Massimo Bagnarol and Fabio Perroni},
     title = {On the {Motivic} {Class} of the {Moduli} {Stack} of {Twisted} $G${-Covers}},
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}
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Massimo Bagnarol; Fabio Perroni. On the Motivic Class of the Moduli Stack of Twisted $G$-Covers. Symmetry, integrability and geometry: methods and applications, Tome 19 (2023). http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a106/

[1] Abramovich D., Corti A., Vistoli A., “Twisted bundles and admissible covers”, Comm. Algebra, 31 (2003), 3547–3618, arXiv: math.AG/0106211 | DOI | MR | Zbl

[2] Abramovich D., Graber T., Vistoli A., “Gromov–Witten theory of Deligne–Mumford stacks”, Amer. J. Math., 130 (2008), 1337–1398, arXiv: math.AG/060315 | DOI | MR | Zbl

[3] Abramovich D., Vistoli A., “Compactifying the space of stable maps”, J. Amer. Math. Soc., 15 (2002), 27–75, arXiv: math.AG/9908167 | DOI | MR | Zbl

[4] Andreini E., Jiang Y., Tseng H.-H., “Gromov–Witten theory of product stacks”, Comm. Anal. Geom., 24 (2016), 223–277, arXiv: 0905.2258 | DOI | MR | Zbl

[5] Arbarello E., Cornalba M., Griffiths P.A., Geometry of algebraic curves, v. II, Grundlehren Math. Wiss., 268, Springer, Heidelberg, 2011 | DOI | MR | Zbl

[6] Artin M., Grothendieck A., Verdier J., Théorie des topos et cohomologie étale des schémas. Séminaire de Géométrie Algébrique du Bois Marie 1963/64 SGA 4, v. 1, Lecture Notes in Math., 269, Springer, Berlin, 1972 | DOI | MR

[7] Bagnarol M., On the cohomology of moduli spaces of stable maps to Grassmannians, Ph.D. Thesis, SISSA, 2019 https://iris.sissa.it/handle/20.500.11767/103198

[8] Bagnarol M., “Betti numbers of stable map spaces to Grassmannians”, Math. Nachr., 295 (2022), 1869–1900, arXiv: 1911.05674 | DOI | MR | Zbl

[9] Behrend K., “Cohomology of stacks”, Intersection Theory and Moduli, ICTP Lect. Notes, 19, Abdus Salam International Centre for Theoretical Physics, Trieste, 2004, 249–294 | MR | Zbl

[10] Behrend K., Dhillon A., “On the motivic class of the stack of bundles”, Adv. Math., 212 (2007), 617–644 | DOI | MR | Zbl

[11] Behrend K., Manin Yu., “Stacks of stable maps and Gromov–Witten invariants”, Duke Math. J., 85 (1996), 1–60, arXiv: alg-geom/9506023 | DOI | MR | Zbl

[12] Brandt M., Chan M., Kannan S., On the weight zero compactly supported cohomology of $\mathcal{H}_{g,n}$, arXiv: 2307.01819

[13] Catanese F., Lönne M., Perroni F., “Irreducibility of the space of dihedral covers of the projective line of a given numerical type”, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 22 (2011), 291–309, arXiv: 1102.0490 | DOI | MR | Zbl

[14] Catanese F., Lönne M., Perroni F., “The irreducible components of the moduli space of dihedral covers of algebraic curves”, Groups Geom. Dyn., 9 (2015), 1185–1229, arXiv: 1206.5498 | DOI | MR | Zbl

[15] Catanese F., Lönne M., Perroni F., “Genus stabilization for the components of moduli spaces of curves with symmetries”, Algebr. Geom., 3 (2016), 23–49, arXiv: 1301.4409 | DOI | MR | Zbl

[16] Chen W., Ruan Y., “Orbifold Gromov–Witten theory”, Orbifolds in Mathematics and Physics (Madison, WI), Contemp. Math., 310, American Mathematical Society, Providence, RI, 2001, 25–85, arXiv: math.AG/0103156 | DOI | MR

[17] Day B., “On closed categories of functors”, Reports of the Midwest Category Seminar, v. IV, Lecture Notes in Math., 137, Springer, Berlin, 1970, 1–38 | DOI | MR

[18] del Baño Rollin S., Navarro Aznar V., “On the motive of a quotient variety”, Collect. Math., 49 (1998) | MR

[19] Deligne P., “Théorie de Hodge. II”, Inst. Hautes Études Sci. Publ. Math., 40 (1971), 5–57 | DOI | MR | Zbl

[20] Deligne P., “Théorie de Hodge. III”, Inst. Hautes Études Sci. Publ. Math., 44 (1974), 5–77 | DOI | MR | Zbl

[21] Drummond-Cole G.C., Hackney P., “Coextension of scalars in operad theory”, Math. Z., 301 (2022), 275–314, arXiv: 1906.12275 | DOI | MR | Zbl

[22] Ekedahl T., The Grothendieck group of algebraic stacks, arXiv: 0903.3143

[23] Fantechi B., Göttsche L., “Orbifold cohomology for global quotients”, Duke Math. J., 117 (2003), 197–227, arXiv: math.AG/0104207 | DOI | MR | Zbl

[24] Florentino C., Silva J., “Hodge–Deligne polynomials of character varieties of free abelian groups”, Open Math., 19 (2021), 338–362, arXiv: 1711.07909 | DOI | MR | Zbl

[25] Fulton W., “Hurwitz schemes and irreducibility of moduli of algebraic curves”, Ann. of Math., 90 (1969), 542–575 | DOI | MR | Zbl

[26] Getzler E., “Operads and moduli spaces of genus $0$ Riemann surfaces”, The Moduli Space of Curves (Texel Island, 1994), Progr. Math., 129, Birkhäuser, Boston, MA, 1995, 199–230, arXiv: alg-geom/9411004 | DOI | MR | Zbl

[27] Getzler E., Pandharipande R., “The Betti numbers of $\overline{\mathcal M}_{0,n}(r,d)$”, J. Algebraic Geom., 15 (2006), 709–732, arXiv: math.AG/0502525 | DOI | MR | Zbl

[28] Gillet H., Soulé C., “Descent, motives and $K$-theory”, J. Reine Angew. Math., 478 (1996), 127–176, arXiv: alg-geom/9507013 | DOI | MR | Zbl

[29] Göttsche L., “On the motive of the Hilbert scheme of points on a surface”, Math. Res. Lett., 8 (2001), 613–627 | DOI | MR | Zbl

[30] Guillén F., Navarro Aznar V., “Un critère d'extension des foncteurs définis sur les schémas lisses”, Publ. Math. Inst. Hautes Études Sci., 95 (2002), 1–91 | DOI | MR | Zbl

[31] Harris J., Mumford D., “On the Kodaira dimension of the moduli space of curves”, Invent. Math., 67 (1982), 23–88 | DOI | MR

[32] Jarvis T.J., Kaufmann R., Kimura T., “Pointed admissible $G$-covers and $G$-equivariant cohomological field theories”, Compos. Math., 141 (2005), 926–978, arXiv: math.AG/0302316 | DOI | MR | Zbl

[33] Joyce D., “Motivic invariants of Artin stacks and ‘stack functions’”, Q. J. Math., 58 (2007), 345–392, arXiv: math.AG/0509722 | DOI | MR | Zbl

[34] Keel S., “Intersection theory of moduli space of stable $n$-pointed curves of genus zero”, Trans. Amer. Math. Soc., 330 (1992), 545–574 | DOI | MR | Zbl

[35] Kelly G.M., Basic concepts of enriched category theory, Repr. Theory Appl. Categ., 10, 2005, vi+137 pp. | MR | Zbl

[36] Kelly G.M., “On the operads of JP May”, Repr. Theory Appl. Categ., 13 (2005), 1–13 | MR | Zbl

[37] Loregian F., (Co)end calculus, London Math. Soc. Lecture Note Ser., 468, Cambridge University Press, Cambridge, 2021 | DOI | MR

[38] Mac Lane S., Categories for the working mathematician, Grad. Texts in Math., 5, Springer, New York, 1998 | DOI | MR | Zbl

[39] Manin Yu.I., “Generating functions in algebraic geometry and sums over trees”, The Moduli Space of Curves (Texel Island, 1994), Progr. Math., 129, Birkhäuser, Boston, MA, 1995, 401–417, arXiv: alg-geom/9407005 | DOI | MR | Zbl

[40] Olsson M.C., “(Log) twisted curves”, Compos. Math., 143 (2007), 476–494 | DOI | MR | Zbl

[41] Peters C., Tata lectures on motivic aspects of Hodge theory, TIFR-Lecture Notes on Math., 92, Tata Institute, Mumbai, 2010

[42] Peters C., Steenbrink J.H.M., Mixed Hodge structures, Ergeb. Math. Grenzgeb. (3), 52, Springer, Berlin, 2008 | DOI | MR | Zbl

[43] Petersen D., “The operad structure of admissible $G$-covers”, Algebra Number Theory, 7 (2013), 1953–1975, arXiv: 1205.0420 | DOI | MR | Zbl

[44] Pikaart M., de Jong A.J., “Moduli of curves with non-abelian level structure”, The Moduli Space of Curves (Texel Island, 1994), Progr. Math., 129, Birkhäuser, Boston, MA, 1995, 483–509, arXiv: alg-geom/9501003 | DOI | MR | Zbl

[45] Romagny M., “Group actions on stacks and applications”, Michigan Math. J., 53 (2005), 209–236 | DOI | MR | Zbl

[46] Toën B., Grothendieck rings of Artin $n$-stacks, arXiv: math.AG/0509098

[47] Voisin C., Hodge theory and complex algebraic geometry, v. I, Cambridge Stud. Adv. Math., 76, Cambridge University Press, Cambridge, 2002 | DOI | MR | Zbl

[48] Yau D., Colored operads, Grad. Stud. Math., 170, American Mathematical Society, Providence, RI, 2016 | DOI | MR | Zbl