Building data for stacky covers

• Eric Ahlqvist ^[1]
  1. [1] University of Edinburgh

     University of Edinburgh

     Reino Unido

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 3, 2024, págs. 1-64
• Idioma: inglés
• DOI: 10.1007/s00029-024-00939-1
• Enlaces
  • Texto completo
• Resumen

  • We define stacky building data for stacky covers in the spirit of Pardini and give an equivalence of (2,1)-categories between the category of stacky covers and the category of stacky building data. We show that every stacky cover is a flat root stack in the sense of Olsson and Borne–Vistoli and give an intrinsic description of it as a root stack using stacky building data. When the base scheme S is defined over a field, we give a criterion for when a birational building datum comes from a tamely ramified cover for a finite abelian group scheme, generalizing a result of Biswas–Borne.

• Referencias bibliográficas
  • Abramovich, D., Graber, T., Vistoli, A.: Gromov-Witten theory of Deligne-Mumford stacks. Am. J. Math. 130(5), 1337–1398 (2008)
  • Altman, A., Kleiman, S.: Introduction to Grothendieck Duality Theory. Lecture Notes in Mathematics, vol. 146. Springer, Berlin (1970)
  • Alper, J.: Good moduli spaces for Artin stacks. ProQuest LLC, Ann Arbor, Thesis (Ph.D.)–Stanford University (2008)
  • Abramovich, D., Olsson, M., Vistoli, A.: Tame stacks in positive characteristic. Ann. Inst. Fourier (Gr.) 58(4), 1057–1091 (2008)
  • Biswas, I., Borne, N.: Tamely ramified Torsors and parabolic bundles. arXiv e-prints arXiv:1706.06733 (2017)
  • Bergh, D.: Functorial destackification of tame stacks with abelian stabilisers. Compos. Math. 153(6), 1257–1315 (2017)
  • Bergh, D., Rydh, D.: Functorial destackification and weak factorization of orbifolds. arXiv:1905.00872 (2019)
  • Brochard, S.: Foncteur de Picard d’un champ algébrique. Math. Ann. 343(3), 541–602 (2009)
  • Borne, N., Vistoli, A.: Parabolic sheaves on logarithmic schemes. Adv. Math. 231(3–4), 1327–1363 (2012)
  • Borne, N., Vistoli, A.: The Nori fundamental Gerbe of a fibered category. J. Algebraic Geom. 24(2), 311–353 (2015)
  • Cadman, C.: Using stacks to impose tangency conditions on curves. Am. J. Math. 129(2), 405–427 (2007)
  • Cadman, C., Chen, L.: Enumeration of rational plane curves tangent to a smooth cubic. Adv. Math. 219(1), 316–343 (2008)
  • Deligne, P.: Le groupe fondamental de la droite projective moins trois points, Galois groups over Q, vol. 16. Mathematical Sciences Research...
  • Eisenbud, D.: Commutative algebra. In: Graduate Texts in Mathematics (With a View Toward Algebraic Geometry), vol. 150. Springer, New York...
  • Evans, K., Konikoff, M., Madden, J.J., Mathis, R., Whipple, G.: Totally ordered commutative monoids. In: Semigroup Forum, vol. 62, no. 2,...
  • Hall, J., Rydh, D.: Coherent Tannaka duality and algebraicity of Hom-stacks. Algebra Number Theory 13(7), 1633–1675 (2019)
  • Inasaridze, H.N.: Extensions of regular semigroups. Soobšˇc. Akad. Nauk Gruzin. SSR 39, 3–10 (1965)
  • Mac Lane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, 2nd edn., vol. 5. Springer, New York (1998)
  • Matsuki, K., Olsson, M.: Kawamata-Viehweg vanishing as Kodaira vanishing for stacks. Math. Res. Lett. 12(2–3), 207–217 (2005)
  • Ogus, A.: Lectures on Logarithmic Algebraic Geometry, Cambridge Studies in Advanced Mathematics, vol. 178. Cambridge University Press, Cambridge...
  • Pardini, R.: Abelian covers of algebraic varieties. J. R. Angew. Math. 417, 191–213 (1991)
  • Patchkoria, A.: The cohomology of monoids with coefficients in semimodules. Sakharth. SSR Mecn. Akad. Moambe 86(3), 545–548 (1977)
  • Patchkoria, A.: Extensions of semimodules by means of monoids, and their cohomology characterization. Sakharth. SSR Mecn. Akad. Moambe 86(1),...
  • Patchkoria, A.: Cohomology monoids of monoids with coefficients in semimodules II. In: Semigroup Forum, vol. 97, no. 1, pp. 131–153 (2018)
  • Rédei, L.: Die Verallgemeinerung der Schreierschen Erweiterungstheorie. Acta Sci. Math. (Szeged) 14, 252–273 (1952)
  • Rydh, D.: Compactification of Tame Deligne–Mumford Stacks. Link
  • Stacks Project Authors: The Stacks Project. Link
  • Strecker, R.: Über kommutative Schreiersche Halbgruppenerweiterungen. Acta Math. Acad. Sci. Hungar. 23, 33–44 (1972)
  • Tonini, F.: The stack of ramified Galois covers. arXiv:1307.1116 (2013)
  • Tonini, F.: Stacks of ramified covers under diagonalizable group schemes. In: International Mathematics Research Notices IMRN, no. 8, pp....
  • William, C.: Waterhouse, Introduction to Affine Group Schemes, Graduate Texts in Mathematics, vol.66. Springer, Berlin (1979)