The birational geometry of Rg,2 and Prym-canonical divisorial strata

• Andrei Bud ^[1]
  1. [1] Institut für Mathematik, Goethe University Frankfurt, Germany Institut für Mathematik, Humboldt-Universität zu Berlin, Germany
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 2, 2024, 31 págs.
• Idioma: inglés
• DOI: 10.1007/s00029-023-00907-1
• Enlaces
  • Texto completo
• Resumen

  • We prove that the moduli space of double covers ramified at two points Rg,2 is uniruled for 3 ≤ g ≤ 6 and of general type for g ≥ 16. Furthermore, we consider Prymcanonical divisorial strata in the moduli space CnRg parametrizing n-pointed Prym curves, and we compute their classes in PicQ(CnRg).

• Referencias bibliográficas
  • Abramovich, D., Corti, A., Vistoli, A.: Twisted bundles and admissible covers. Commun. Algebra 8, 3547–3618 (2003)
  • Aprodu, M., Farkas, G.: Green’s conjecture for general covers. Contemp. Math. 564, 211–226 (2012)
  • Arbarello, E., Cornalba, M.: The Picard groups of the moduli spaces of curves. Topology 26, 153–171 (1987)
  • Arbarello, E., Cornalba, M.: Calculating cohomology groups of moduli spaces of curves via algebraic geometry. Publ. Math. de l’Inst. des Hautes...
  • Bainbridge, M., Chen, D., Gendron, Q., Grushevsky, S., Möller, M.: Strata of k-differentials. Algebr. Geom. 6, 196–233 (2019)
  • Ballico, E., Casagrande, C., Fontanari, C.: Moduli of Prym curves. Doc. Math. 9, 265–281 (2004)
  • Barros, I.: Uniruledness of strata of holomorphic differentials in small genus. Adv. Math. 333, 670–693 (2018)
  • Beauville, A.: Variétés de Prym et jacobiennes intermédiaires. Ann. Sci. de l’École Normale Supérieure, 4e série 10, 309–391 (1977)
  • Bernstein, M.: Moduli of curves with level structures. Harvard University Ph.D. Thesis (1999)
  • Bertram, A.: An existence theorem for Prym special divisors. Invent. Math. 90, 669–671 (1987)
  • Bruns, G.: R15 is of general type. Algebra Number Theory 10, 1949–1964 (2016)
  • Bud, A.: A Hurwitz divisor on the moduli of Prym curves. Preprint, arXiv:2104.14947 (2021)
  • Cartan, H.: Quotient d’une variété analytique par un groupe discret d’automorphismes. Séminaire Henri Cartan, 6, 1953–1954. talk:12
  • Catanese, F.: On the Rationality of Certain Moduli Spaces Related to Curves of Genus 4. Springer Lecture Notes in Mathematics 1008, 30–50...
  • Cornalba, M.: Moduli of curves and theta-characteristics. In: Lectures on Riemann surfaces (Trieste, 1987), pp. 560–589. World Scientific...
  • Diaz, S.: Exceptional Weierstrass Points and the Divisor on Moduli Space that they Define, vol. 56. Memoirs of the American Mathematical Society...
  • Dolgachev, I.: Rationality of fields of invariants. Algebraic Geometry Bowdoin 1985, Proceedings of Symphosia in Pure Mathematics 46(Part...
  • Donagi, R.: The fibers of the Prym map. Contemp. Math. 136, 55–125 (1982)
  • Donagi, R.: The unirationality of A5. Ann. Math. 119, 269–307 (1984)
  • Donagi, R., Smith, R.: The structure of the Prym map. Acta Math. 146, 25–102 (1981)
  • Eisenbud, D., Harris, J.: A simpler proof of the Gieseker-Petri Theorem on special divisors. Invent. Math. 74, 269–280 (1983)
  • Eisenbud, D., Harris, J.: Limit linear series: basic theory. Invent. Math. 85, 337–372 (1986)
  • Eisenbud, D., Harris, J.: The Kodaira dimension of the moduli space of curves of genus ≥ 23. Invent. Math. 90, 359–387 (1987)
  • Farkas, G.: Koszul divisors on moduli spaces of curves. Am. J. Math. 131, 819–867 (2009)
  • Farkas, G., Jensen, D., Payne, S.: The Kodaira dimensions of M22 and M23. Preprint, arXiv:2005.00622 (2020)
  • Farkas, G., Ludwig, K.: The Kodaira dimension of the moduli space of Prym varieties. J. Eur. Math. Soc. 12, 755–795 (2010)
  • Farkas, G., Verra, A.: The classification of universal Jacobians over the moduli space of curves. Commentarii Math. Helvetici 88, 587–611...
  • Farkas, G., Verra, A.: The universal theta divisor over the moduli space of curves. J. de Math. Pures et Appl. 100, 591–605 (2013)
  • Farkas, G., Verra, A.: Prym varieties and moduli of polarized Nikulin surfaces. Adv. Math. 290, 314–328 (2016)
  • Grushevsky, S., Zakharov, D.: The double ramification cycle and the theta divisor. Proc. Am. Math. Soc. 142, 4053–4064 (2014)
  • Harris, J., Mumford, D.: On the Kodaira dimension of the moduli space of curves. Invent. Math. 67, 23–88 (1982)
  • Izadi, E., Giudice, M.L., Sankaran, G.: The moduli space of étale double covers of genus 5 curves is unirational. Pac. J. Math. 239, 39–52...
  • Lange, H., Ortega, A.: Prym varieties of triple coverings. Int. Math. Res. Not. 5045–5075, 2009 (2011)
  • Lanneau, E.: Connected components of the strata of the moduli spaces of quadratic differentials. Ann. Sci. de l’École Normale Supérieure 41,...
  • Logan, A.: The Kodaira dimension of moduli spaces of curves with marked points. Am. J. Math. 125, 105–138 (2003)
  • Ludwig, K.: On the geometry of the moduli space of spin curves. J. Algebraic Geom. 19, 133–171 (2010)
  • Maestro Pérez, C.: Prym curves with a vanishing theta null. Preprint, arXiv:2102.03435 (2021)
  • Marcucci, V.O., Pirola, G.P.: Generic Torelli theorem for Prym varieties of ramified coverings. Compos. Math. 148, 1147–1170 (2012)
  • Mori, S., Mukai, S.: The uniruledness of the moduli space of curves of genus 11. Springer Lecture Notes in Mathematics 1016, 334–353 (1983)
  • Mullane, S.: Effective divisors in Mg,n from Abelian differentials. Mich. Math. J. 67, 839–889 (2018)
  • Mullane, S.: k-differentials on curves and rigid cycles in moduli space. Preprint, arXiv:1905.03241 (2019)
  • Mumford, D.: Prym varieties I. Contributions to analysis, pp. 325–350 (1974)
  • Müller, F.: The pullback of a theta divisor to Mg,n. Math. Nachr. 286, 1255–1266 (2013)
  • Naranjo, J., Ortega, A.: Global Prym–Torelli for double coverings ramified in at least 6 points. Preprint, arXiv:2005.11108, to appear in...
  • Naranjo, J., Ortega, A., Verra, A.: Generic injectivity of the Prym map for double ramified coverings. Trans. Am. Math. Soc. 371, 3627–3646...
  • Prill, D.: Local classification of quotients of complex manifolds by discontinuous groups. Duke Math. J. 34, 375–386 (1967)
  • Schwarz, I.: On the Kodaira dimension of the moduli space of nodal curves. Annali della Scuola Normale di Pisa 22, 463–484 (2021)
  • Stanley, R.P.: Catalan Numbers. Cambridge University Press, Cambridge (2015)
  • Ueno, K.: Classification theory of algebraic varieties and compact complex spaces, vol. 439. Lecture Notes in Mathematics, Springer, Berlin...
  • Verra, A.: A short proof of the unirationality of A5. Indagationes Math. (Proceedings) 87, 339–355 (1984)
  • Verra, A.: On the universal principally polarized abelian variety of dimension 4. In: Curves and abelian varieties (Athens, Georgia, 2007)...
  • Viehweg, E.: Die additivität der Kodaira dimension für projektive Faserräume über Varietäten des allgemeinen Typs. J. für die reine und Angew....