## Moduli of curves and spin structures via algebraic geometry

HTML articles powered by AMS MathViewer

- by Gilberto Bini and Claudio Fontanari PDF
- Trans. Amer. Math. Soc.
**358**(2006), 3207-3217 Request permission

## Abstract:

Here we investigate some birational properties of two collections of moduli spaces, namely moduli spaces of (pointed) stable curves and of (pointed) spin curves. In particular, we focus on vanishings of Hodge numbers of type $(p,0)$ and on computations of Kodaira dimension. Our methods are purely algebro-geometric and rely on an induction argument on the number of marked points and the genus of the curves.## References

- Dan Abramovich and Tyler J. Jarvis,
*Moduli of twisted spin curves*, Proc. Amer. Math. Soc.**131**(2003), no. 3, 685–699. MR**1937405**, DOI 10.1090/S0002-9939-02-06562-0 - Enrico Arbarello and Maurizio Cornalba,
*The Picard groups of the moduli spaces of curves*, Topology**26**(1987), no. 2, 153–171. MR**895568**, DOI 10.1016/0040-9383(87)90056-5 - Enrico Arbarello and Maurizio Cornalba,
*Calculating cohomology groups of moduli spaces of curves via algebraic geometry*, Inst. Hautes Études Sci. Publ. Math.**88**(1998), 97–127 (1999). MR**1733327**, DOI 10.1007/BF02701767 - Enrico Arbarello and Maurizio Cornalba,
*Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves*, J. Algebraic Geom.**5**(1996), no. 4, 705–749. MR**1486986** - Belorousski, P.:
*Chow rings of moduli spaces of pointed elliptic curves*. Ph.D. thesis, Chicago, 1998. - Gilberto Bini,
*Generalized Hodge classes on the moduli space of curves*, Beiträge Algebra Geom.**44**(2003), no. 2, 559–565. MR**2017857** - Lucia Caporaso and Cinzia Casagrande,
*Combinatorial properties of stable spin curves*, Comm. Algebra**31**(2003), no. 8, 3653–3672. Special issue in honor of Steven L. Kleiman. MR**2007379**, DOI 10.1081/AGB-120022437 - Maurizio Cornalba,
*Moduli of curves and theta-characteristics*, Lectures on Riemann surfaces (Trieste, 1987) World Sci. Publ., Teaneck, NJ, 1989, pp. 560–589. MR**1082361** - Maurizio Cornalba,
*A remark on the Picard group of spin moduli space*, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.**2**(1991), no. 3, 211–217 (English, with Italian summary). MR**1135424** - Maurizio Cornalba,
*Cohomology of moduli spaces of stable curves*, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 249–257. MR**1648075** - Carel Faber and Eduard Looijenga,
*Remarks on moduli of curves*, Moduli of curves and abelian varieties, Aspects Math., E33, Friedr. Vieweg, Braunschweig, 1999, pp. 23–45. MR**1722537** - Gavril Farkas,
*The geometry of the moduli space of curves of genus 23*, Math. Ann.**318**(2000), no. 1, 43–65. MR**1785575**, DOI 10.1007/s002080000108 - E. Getzler,
*The semi-classical approximation for modular operads*, Comm. Math. Phys.**194**(1998), no. 2, 481–492. MR**1627677**, DOI 10.1007/s002200050365 - Getzler, E.: The Hodge polynomials of ${\overline {\mathcal M}}_{3,1}$. Preprint http://xxx.lanl.gov/abs/ math.AG/9910174 (1999).
- T. Graber and R. Pandharipande,
*Constructions of nontautological classes on moduli spaces of curves*, Michigan Math. J.**51**(2003), no. 1, 93–109. MR**1960923**, DOI 10.1307/mmj/1049832895 - Phillip Griffiths and Joseph Harris,
*Principles of algebraic geometry*, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR**507725** - John Harer,
*The second homology group of the mapping class group of an orientable surface*, Invent. Math.**72**(1983), no. 2, 221–239. MR**700769**, DOI 10.1007/BF01389321 - John L. Harer,
*The virtual cohomological dimension of the mapping class group of an orientable surface*, Invent. Math.**84**(1986), no. 1, 157–176. MR**830043**, DOI 10.1007/BF01388737 - John L. Harer,
*Stability of the homology of the moduli spaces of Riemann surfaces with spin structure*, Math. Ann.**287**(1990), no. 2, 323–334. MR**1054572**, DOI 10.1007/BF01446896 - John L. Harer,
*The rational Picard group of the moduli space of Riemann surfaces with spin structure*, Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991) Contemp. Math., vol. 150, Amer. Math. Soc., Providence, RI, 1993, pp. 107–136. MR**1234262**, DOI 10.1090/conm/150/01288 - Joe Harris and Ian Morrison,
*Moduli of curves*, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, New York, 1998. MR**1631825** - Joe Harris and David Mumford,
*On the Kodaira dimension of the moduli space of curves*, Invent. Math.**67**(1982), no. 1, 23–88. With an appendix by William Fulton. MR**664324**, DOI 10.1007/BF01393371 - Tyler J. Jarvis,
*Geometry of the moduli of higher spin curves*, Internat. J. Math.**11**(2000), no. 5, 637–663. MR**1780734**, DOI 10.1142/S0129167X00000325 - Tyler J. Jarvis,
*The Picard group of the moduli of higher spin curves*, New York J. Math.**7**(2001), 23–47. MR**1838471** - Sean Keel,
*Intersection theory of moduli space of stable $n$-pointed curves of genus zero*, Trans. Amer. Math. Soc.**330**(1992), no. 2, 545–574. MR**1034665**, DOI 10.1090/S0002-9947-1992-1034665-0 - Adam Logan,
*The Kodaira dimension of moduli spaces of curves with marked points*, Amer. J. Math.**125**(2003), no. 1, 105–138. MR**1953519**, DOI 10.1353/ajm.2003.0005 - Martin Pikaart,
*An orbifold partition of $\overline M{}^n_g$*, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 467–482. MR**1363067**, DOI 10.1007/978-1-4612-4264-2_{1}7 - Kenji Ueno,
*Classification theory of algebraic varieties and compact complex spaces*, Lecture Notes in Mathematics, Vol. 439, Springer-Verlag, Berlin-New York, 1975. Notes written in collaboration with P. Cherenack. MR**0506253**, DOI 10.1007/BFb0070570

## Additional Information

**Gilberto Bini**- Affiliation: Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy
- Email: gilberto.bini@mat.unimi.it
**Claudio Fontanari**- Affiliation: Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
- Email: fontanar@science.unitn.it
- Received by editor(s): January 13, 2004
- Received by editor(s) in revised form: September 8, 2004
- Published electronically: February 20, 2006
- © Copyright 2006 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**358**(2006), 3207-3217 - MSC (2000): Primary 14H10, 14E08
- DOI: https://doi.org/10.1090/S0002-9947-06-03838-4
- MathSciNet review: 2216264