Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Effective divisors on $\overline{\mathcal{M}}_g$, curves on $K3$ surfaces, and the slope conjecture


Authors: Gavril Farkas and Mihnea Popa
Journal: J. Algebraic Geom. 14 (2005), 241-267
DOI: https://doi.org/10.1090/S1056-3911-04-00392-3
Published electronically: November 18, 2004
MathSciNet review: 2123229
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Abstract | References | Additional Information

Abstract: We compute the class of the compactification of the divisor of curves sitting on a $K3$ surface and show that it violates the Harris-Morrison Slope Conjecture. We carry this out using the fact that this divisor has four distinct incarnations as a geometric subvariety of the moduli space of curves. We also give a counterexample to a hypothesis raised by Harris and Morrison that the Brill-Noether divisors are essentially the only effective divisors on the moduli space of curves having minimal slope $6+12/(g+1)$.


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Additional Information

Gavril Farkas
Affiliation: Department of Mathematics, Princeton University, Fine Hall, Princeton, New Jersey 08544
Address at time of publication: Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, Texas 78712
Email: gfarkas@math.princeton.edu

Mihnea Popa
Affiliation: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138
Email: mpopa@math.harvard.edu

DOI: https://doi.org/10.1090/S1056-3911-04-00392-3
Received by editor(s): May 16, 2003
Published electronically: November 18, 2004
Additional Notes: The first author’s research was partially supported by NSF Grant DMS-0140520. The second author’s research was partially supported by NSF Grant DMS-0200150

American Mathematical Society