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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

Author(s): Gavril Farkas; Mihnea Popa
Journal: J. Algebraic Geom. 14 (2005), 241-267.
Posted: November 18, 2004
MathSciNet review: 2123229
Retrieve article in: PDF

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: 10.1090/S1056-3911-04-00392-3
PII: S 1056-3911(04)00392-3
Received by editor(s): May 16, 2003
Posted: 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


Journal of Algebraic Geometry
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