Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Rational maps between moduli spaces of curves and Gieseker-Petri divisors


Author: Gavril Farkas
Journal: J. Algebraic Geom. 19 (2010), 243-284
DOI: https://doi.org/10.1090/S1056-3911-09-00510-4
Published electronically: June 2, 2009
MathSciNet review: 2580676
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Abstract: We study contractions of the moduli space of stable curves beyond the minimal model of $ \overline{\mathcal{M}}_{g'}$ by giving a complete enumerative description of the rational map between two moduli spaces of curves $ \overline{\mathcal{M}}_g \dashrightarrow\overline{\mathcal{M}}_{g'}$ which associates to a curve $ C$ of genus $ g$ its Brill-Noether locus of special divisors in the case this locus is a curve. As an application we construct many examples of moving effective divisors on $ \overline{\mathcal{M}}_g$ of small slope, which in turn can be used to show that various moduli space of curves with level structure are of general type. For low $ g'$ our calculation can be used to study the intersection theory of the moduli space of Prym varieties of dimension $ 5$.


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Gavril Farkas
Affiliation: Humboldt-Universität zu Berlin, Institut für Mathematik, 10099 Berlin, Germany
Email: farkas@math.hu-berlin.de

DOI: https://doi.org/10.1090/S1056-3911-09-00510-4
Received by editor(s): September 19, 2007
Received by editor(s) in revised form: December 19, 2007
Published electronically: June 2, 2009
Additional Notes: Research was partially supported by an Alfred P. Sloan Fellowship, the NSF Grants DMS-0450670 and DMS-0500747 and a 2006 Texas Summer Research Assignment. Most of this paper was written while visiting the Institut Mittag-Leffler in Djursholm in the Spring of 2007. Support from the institute is gratefully acknowledged.

American Mathematical Society