The geometry of the moduli space of odd spin curves

Abstract

The spin moduli space $\overline{\mathcal{S}}_g$ is the parameter space of theta characteristics (spin structures) on stable curves of genus $g$. It has two connected components, $\overline{\mathcal{S}}_g^-$ and $\overline{\mathcal{S}}_g^+$, depending on the parity of the spin structure. We establish a complete birational classification by Kodaira dimension of the odd component $\overline{\mathcal{S}}_g^-$ of the spin moduli space. We show that $\overline{\mathcal{S}}_g^-$ is uniruled for $g<12$ and even unirational for $g\leq 8$. In this range, introducing the concept of cluster for the Mukai variety whose one-dimensional linear sections are general canonical curves of genus $g$, we construct new birational models of $\overline{\mathcal{S}}_g^-$. These we then use to explicitly describe the birational structure of $\overline{\mathcal{S}}_g^-$. For instance, $\overline{\mathcal{S}}_8^-$ is birational to a locally trivial $\textbf{P}^7$-bundle over the moduli space of elliptic curves with seven pairs of marked points. For $g\geq 12$, we prove that $\overline{\mathcal{S}}_g^-$ is a variety of general type. In genus $12$, this requires the construction of a counterexample to the Slope Conjecture on effective divisors on the moduli space of stable curves of genus $12$.

  • [Dol] Go to document I. V. Dolgachev, Classical Algebraic Geometry: A Modern View, Cambridge: Cambridge Univ. Press, 2012.
    @book{Dol, address = {Cambridge},
      author = {Dolgachev, Igor V.},
      pages = {xii+639},
      publisher = {Cambridge Univ. Press},
      title = {Classical {A}lgebraic {G}eometry: A {M}odern {V}iew},
      year = {2012},
      doi = {10.1017/CBO9781139084437},
      isbn = {978-1-107-01765-8},
      }
  • [DK] Go to document I. V. Dolgachev and V. Kanev, "Polar covariants of plane cubics and quartics," Adv. Math., vol. 98, iss. 2, pp. 216-301, 1993.
    @article{DK,
      author = {Dolgachev, Igor V. and Kanev, Vassil},
      journal = {Adv. Math.},
      number = {2},
      pages = {216--301},
      title = {Polar covariants of plane cubics and quartics},
      volume = {98},
      year = {1993},
      doi = {10.1006/aima.1993.1016},
      issn = {0001-8708},
      }
  • [CS] Go to document L. Caporaso and E. Sernesi, "Characterizing curves by their odd theta-characteristics," J. Reine Angew. Math., vol. 562, pp. 101-135, 2003.
    @article{CS,
      author = {Caporaso, Lucia and Sernesi, Edoardo},
      journal = {J. Reine Angew. Math.},
      pages = {101--135},
      title = {Characterizing curves by their odd theta-characteristics},
      volume = {562},
      year = {2003},
      doi = {10.1515/crll.2003.070},
      issn = {0075-4102},
      }
  • [C] M. Cornalba, "Moduli of curves and theta-characteristics," in Lectures on Riemann Surfaces, Teaneck, NJ: World Sci. Publ., 1989, pp. 560-589.
    @incollection{C, address = {Teaneck, NJ},
      author = {Cornalba, Maurizio},
      booktitle = {Lectures on {R}iemann Surfaces},
      pages = {560--589},
      publisher = {World Sci. Publ.},
      title = {Moduli of curves and theta-characteristics},
      year = {1989},
      }
  • [CCC] Go to document L. Caporaso, C. Casagrande, and M. Cornalba, "Moduli of roots of line bundles on curves," Trans. Amer. Math. Soc., vol. 359, iss. 8, pp. 3733-3768, 2007.
    @article{CCC,
      author = {Caporaso, Lucia and Casagrande, Cinzia and Cornalba, Maurizio},
      journal = {Trans. Amer. Math. Soc.},
      number = {8},
      pages = {3733--3768},
      title = {Moduli of roots of line bundles on curves},
      volume = {359},
      year = {2007},
      doi = {10.1090/S0002-9947-07-04087-1},
      issn = {0002-9947},
      }
  • [AJ] Go to document D. Abramovich and T. J. Jarvis, "Moduli of twisted spin curves," Proc. Amer. Math. Soc., vol. 131, iss. 3, pp. 685-699, 2003.
    @article{AJ,
      author = {Abramovich, Dan and Jarvis, Tyler J.},
      journal = {Proc. Amer. Math. Soc.},
      number = {3},
      pages = {685--699},
      title = {Moduli of twisted spin curves},
      volume = {131},
      year = {2003},
      doi = {10.1090/S0002-9939-02-06562-0},
      issn = {0002-9939},
      }
  • [FP] Go to document G. Farkas and M. Popa, "Effective divisors on $\overline{\mathcal M}_g$, curves on $K3$ surfaces, and the slope conjecture," J. Algebraic Geom., vol. 14, iss. 2, pp. 241-267, 2005.
    @article{FP,
      author = {Farkas, Gavril and Popa, Mihnea},
      journal = {J. Algebraic Geom.},
      number = {2},
      pages = {241--267},
      title = {Effective divisors on {$\overline{\mathcal M}\sb g$},
      curves on {$K3$} surfaces, and the slope conjecture},
      volume = {14},
      year = {2005},
      doi = {10.1090/S1056-3911-04-00392-3},
      issn = {1056-3911},
      }
  • [M1] S. Mukai, "Curves and Grassmannians," in Algebraic Geometry and Related Topics, Yang, J. -H., Namikawa, Y., and Ueno, K., Eds., Int. Press, Cambridge, MA, 1993, vol. I, pp. 19-40.
    @incollection{M1,
      author = {Mukai, Shigeru},
      booktitle = {Algebraic Geometry and Related Topics},
      editor = {Yang, J.-H. and Namikawa, Y. and Ueno, K.},
      pages = {19--40},
      publisher = {Int. Press, Cambridge, MA},
      series = {Conf. Proc. Lecture Notes Algebraic Geom.},
      title = {Curves and {G}rassmannians},
      volume = {I},
      year = {1993},
      }
  • [M2] Go to document S. Mukai, "Curves and symmetric spaces. I," Amer. J. Math., vol. 117, iss. 6, pp. 1627-1644, 1995.
    @article{M2,
      author = {Mukai, Shigeru},
      journal = {Amer. J. Math.},
      number = {6},
      pages = {1627--1644},
      title = {Curves and symmetric spaces. {I}},
      volume = {117},
      year = {1995},
      doi = {10.2307/2375032},
      issn = {0002-9327},
      }
  • [M3] Go to document S. Mukai, "Curves and symmetric spaces, II," Ann. of Math., vol. 172, iss. 3, pp. 1539-1558, 2010.
    @article{M3,
      author = {Mukai, Shigeru},
      journal = {Ann. of Math.},
      number = {3},
      pages = {1539--1558},
      title = {Curves and symmetric spaces, {II}},
      volume = {172},
      year = {2010},
      doi = {10.4007/annals.2010.172.1539},
      issn = {0003-486X},
      }
  • [F3] Go to document G. Farkas, "The birational type of the moduli space of even spin curves," Adv. Math., vol. 223, iss. 2, pp. 433-443, 2010.
    @article{F3,
      author = {Farkas, Gavril},
      journal = {Adv. Math.},
      number = {2},
      pages = {433--443},
      title = {The birational type of the moduli space of even spin curves},
      volume = {223},
      year = {2010},
      doi = {10.1016/j.aim.2009.08.011},
      issn = {0001-8708},
      }
  • [FV] Go to document G. Farkas and A. Verra, "Moduli of theta-characteristics via Nikulin surfaces," Math. Ann., vol. 354, iss. 2, pp. 465-496, 2012.
    @article{FV,
      author = {Farkas, Gavril and Verra, Alessandro},
      journal = {Math. Ann.},
      number = {2},
      pages = {465--496},
      title = {Moduli of theta-characteristics via {N}ikulin surfaces},
      volume = {354},
      year = {2012},
      doi = {10.1007/s00208-011-0739-z},
      issn = {0025-5831},
      }
  • [Lud] Go to document K. Ludwig, "On the geometry of the moduli space of spin curves," J. Algebraic Geom., vol. 19, iss. 1, pp. 133-171, 2010.
    @article{Lud,
      author = {Ludwig, Katharina},
      journal = {J. Algebraic Geom.},
      number = {1},
      pages = {133--171},
      title = {On the geometry of the moduli space of spin curves},
      volume = {19},
      year = {2010},
      doi = {10.1090/S1056-3911-09-00505-0},
      issn = {1056-3911},
      }
  • [Cu] Go to document F. Cukierman, "Families of Weierstrass points," Duke Math. J., vol. 58, iss. 2, pp. 317-346, 1989.
    @article{Cu,
      author = {Cukierman, Fernando},
      journal = {Duke Math. J.},
      number = {2},
      pages = {317--346},
      title = {Families of {W}eierstrass points},
      volume = {58},
      year = {1989},
      doi = {10.1215/S0012-7094-89-05815-8},
      issn = {0012-7094},
      }
  • [EH2] Go to document D. Eisenbud and J. Harris, "The Kodaira dimension of the moduli space of curves of genus $\geq 23$," Invent. Math., vol. 90, iss. 2, pp. 359-387, 1987.
    @article{EH2,
      author = {Eisenbud, David and Harris, Joe},
      journal = {Invent. Math.},
      number = {2},
      pages = {359--387},
      title = {The {K}odaira dimension of the moduli space of curves of genus {$\geq 23$}},
      volume = {90},
      year = {1987},
      doi = {10.1007/BF01388710},
      issn = {0020-9910},
      }
  • [FL] Go to document G. Farkas and K. Ludwig, "The Kodaira dimension of the moduli space of Prym varieties," J. Eur. Math. Soc. $($JEMS$)$, vol. 12, iss. 3, pp. 755-795, 2010.
    @article{FL,
      author = {Farkas, Gavril and Ludwig, Katharina},
      journal = {J. Eur. Math. Soc. $($JEMS$)$},
      number = {3},
      pages = {755--795},
      title = {The {K}odaira dimension of the moduli space of {P}rym varieties},
      volume = {12},
      year = {2010},
      doi = {10.4171/JEMS/214},
      issn = {1435-9855},
      }
  • [Mu] Go to document D. Mumford, "Theta characteristics of an algebraic curve," Ann. Sci. École Norm. Sup., vol. 4, pp. 181-192, 1971.
    @article{Mu,
      author = {Mumford, David},
      journal = {Ann. Sci. École Norm. Sup.},
      pages = {181--192},
      title = {Theta characteristics of an algebraic curve},
      volume = {4},
      year = {1971},
      issn = {0012-9593},
      url = {http://www.numdam.org/item?id=ASENS_1971_4_4_2_181_0},
      }
  • [HM] Go to document J. Harris and D. Mumford, "On the Kodaira dimension of the moduli space of curves," Invent. Math., vol. 67, iss. 1, pp. 23-88, 1982.
    @article{HM,
      author = {Harris, Joe and Mumford, David},
      journal = {Invent. Math.},
      number = {1},
      pages = {23--88},
      title = {On the {K}odaira dimension of the moduli space of curves},
      volume = {67},
      year = {1982},
      doi = {10.1007/BF01393371},
      issn = {0020-9910},
      }
  • [M4] S. Mukai, "Curves and $K3$ surfaces of genus eleven," in Moduli of Vector Bundles, New York: Dekker, 1996, vol. 179, pp. 189-197.
    @incollection{M4, address = {New York},
      author = {Mukai, Shigeru},
      booktitle = {Moduli of Vector Bundles},
      pages = {189--197},
      publisher = {Dekker},
      series = {Lecture Notes Pure Appl. Math.},
      title = {Curves and {$K3$} surfaces of genus eleven},
      volume = {179},
      year = {1996},
      }
  • [CD] Go to document C. Ciliberto and T. Dedieu, "On universal Severi varieties of low genus $K3$ surfaces," Math. Z., vol. 271, iss. 3-4, pp. 953-960, 2012.
    @article{CD,
      author = {Ciliberto, Ciro and Dedieu, Thomas},
      journal = {Math. Z.},
      number = {3-4},
      pages = {953--960},
      title = {On universal {S}everi varieties of low genus {$K3$} surfaces},
      volume = {271},
      year = {2012},
      doi = {10.1007/s00209-011-0898-3},
      issn = {0025-5874},
      }
  • [La] Go to document R. Lazarsfeld, "Brill-Noether-Petri without degenerations," J. Differential Geom., vol. 23, iss. 3, pp. 299-307, 1986.
    @article{La,
      author = {Lazarsfeld, Robert},
      journal = {J. Differential Geom.},
      number = {3},
      pages = {299--307},
      title = {Brill-{N}oether-{P}etri without degenerations},
      volume = {23},
      year = {1986},
      issn = {0022-040X},
      url = {http://projecteuclid.org/euclid.jdg/1214440116},
      }
  • [Ta] Go to document A. Tannenbaum, "Families of curves with nodes on $K3$ surfaces," Math. Ann., vol. 260, iss. 2, pp. 239-253, 1982.
    @article{Ta,
      author = {Tannenbaum, Allen},
      journal = {Math. Ann.},
      number = {2},
      pages = {239--253},
      title = {Families of curves with nodes on {$K3$} surfaces},
      volume = {260},
      year = {1982},
      doi = {10.1007/BF01457238},
      issn = {0025-5831},
      }
  • [HH] Go to document R. Hartshorne and A. Hirschowitz, "Smoothing algebraic space curves," in Algebraic Geometry, Sitges, New York: Springer-Verlag, 1985, vol. 1124, pp. 98-131.
    @incollection{HH, address = {New York},
      author = {Hartshorne, R. and Hirschowitz, A.},
      booktitle = {Algebraic Geometry, {S}itges},
      pages = {98--131},
      publisher = {Springer-Verlag},
      series = {Lecture Notes in Math.},
      title = {Smoothing algebraic space curves},
      volume = {1124},
      year = {1985},
      doi = {10.1007/BFb0074998},
      }
  • [Sc] G. Scorza, "Sopra le curve canoniche di uno spazio lineare qualunque e sopra certi loro covarianti quartici," Atti Accad. Reale Sci. Torino, vol. 35, pp. 765-773, 1900.
    @article{Sc,
      author = {Scorza, G.},
      journal = {Atti Accad. Reale Sci. Torino},
      pages = {765--773},
      title = {Sopra le curve canoniche di uno spazio lineare qualunque e sopra certi loro covarianti quartici},
      volume = {35},
      year = {1900},
      }
  • [TZ] Go to document H. Takagi and F. Zucconi, "Spin curves and Scorza quartics," Math. Ann., vol. 349, iss. 3, pp. 623-645, 2011.
    @article{TZ,
      author = {Takagi, Hiromichi and Zucconi, Francesco},
      journal = {Math. Ann.},
      number = {3},
      pages = {623--645},
      title = {Spin curves and {S}corza quartics},
      volume = {349},
      year = {2011},
      doi = {10.1007/s00208-010-0530-6},
      issn = {0025-5831},
      }
  • [EH1] Go to document D. Eisenbud and J. Harris, "Limit linear series: Basic theory," Invent. Math., vol. 85, iss. 2, pp. 337-371, 1986.
    @article{EH1,
      author = {Eisenbud, David and Harris, Joe},
      journal = {Invent. Math.},
      number = {2},
      pages = {337--371},
      title = {Limit linear series: {B}asic theory},
      volume = {85},
      year = {1986},
      doi = {10.1007/BF01389094},
      issn = {0020-9910},
      }
  • [F2] Go to document G. Farkas, "Koszul divisors on moduli spaces of curves," Amer. J. Math., vol. 131, iss. 3, pp. 819-867, 2009.
    @article{F2,
      author = {Farkas, Gavril},
      journal = {Amer. J. Math.},
      number = {3},
      pages = {819--867},
      title = {Koszul divisors on moduli spaces of curves},
      volume = {131},
      year = {2009},
      doi = {10.1353/ajm.0.0053},
      issn = {0002-9327},
      }
  • [Log] Go to document A. Logan, "The Kodaira dimension of moduli spaces of curves with marked points," Amer. J. Math., vol. 125, iss. 1, pp. 105-138, 2003.
    @article{Log,
      author = {Logan, Adam},
      journal = {Amer. J. Math.},
      number = {1},
      pages = {105--138},
      title = {The {K}odaira dimension of moduli spaces of curves with marked points},
      volume = {125},
      year = {2003},
      doi = {10.1353/ajm.2003.0005},
      issn = {0002-9327},
      }
  • [F1] Go to document G. Farkas, "Syzygies of curves and the effective cone of $\overline{\mathcal{M}}_g$," Duke Math. J., vol. 135, iss. 1, pp. 53-98, 2006.
    @article{F1,
      author = {Farkas, Gavril},
      journal = {Duke Math. J.},
      number = {1},
      pages = {53--98},
      title = {Syzygies of curves and the effective cone of {$\overline{\mathcal{M}}_g$}},
      volume = {135},
      year = {2006},
      doi = {10.1215/S0012-7094-06-13512-3},
      issn = {0012-7094},
      }
  • [HT] Go to document J. Harris and L. Tu, "Chern numbers of kernel and cokernel bundles," Invent. Math., vol. 75, iss. 3, pp. 467-475, 1984.
    @article{HT,
      author = {Harris, J. and Tu, L.},
      journal = {Invent. Math.},
      number = {3},
      pages = {467--475},
      title = {Chern numbers of kernel and cokernel bundles},
      volume = {75},
      year = {1984},
      doi = {10.1007/BF01388639},
      issn = {0020-9910},
      }
  • [T] Go to document M. Teixidor i Bigas, "Petri map for rank two bundles with canonical determinant," Compos. Math., vol. 144, iss. 3, pp. 705-720, 2008.
    @article{T,
      author = {{Teixidor i Bigas},
      Montserrat},
      journal = {Compos. Math.},
      number = {3},
      pages = {705--720},
      title = {Petri map for rank two bundles with canonical determinant},
      volume = {144},
      year = {2008},
      doi = {10.1112/S0010437X07003442},
      issn = {0010-437X},
      }
  • [Vo] Go to document C. Voisin, "Sur l’application de Wahl des courbes satisfaisant la condition de Brill-Noether-Petri," Acta Math., vol. 168, iss. 3-4, pp. 249-272, 1992.
    @article{Vo,
      author = {Voisin, Claire},
      journal = {Acta Math.},
      number = {3-4},
      pages = {249--272},
      title = {Sur l'application de {W}ahl des courbes satisfaisant la condition de {B}rill-{N}oether-{P}etri},
      volume = {168},
      year = {1992},
      doi = {10.1007/BF02392980},
      issn = {0001-5962},
      }
  • [FR] G. Farkas and R. Rimányi, Symmetric degeneracy loci and moduli of curves.
    @misc{FR,
      author = {Farkas, Gavril and Rim{\'{a}}nyi, R.},
      note = {in preparation},
      title = {Symmetric degeneracy loci and moduli of curves},
      }

Authors

Gavril Farkas

Humboldt-Universität zu Berlin, Berlin, Germany

Alessandro Verra

Universitá Roma Tre, Roma, Italy