Effective divisors on , curves on 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 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 .

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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