Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Completion of Katz-Qin-Ruan's enumeration of genus-two plane curves

Author: Aleksey Zinger
Translated by:
Journal: J. Algebraic Geom. 13 (2004), 547-561
Published electronically: December 8, 2003
MathSciNet review: 2047680
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Abstract | References | Additional Information

Abstract: We give a formula for the number of plane curves of degree $d$ and genus $2$with fixed complex structure passing through $3d-2$ points in general position. This is achieved by completing the Katz-Qin-Ruan approach. This paper's formula agrees with the one obtained by the author in a completely different way.

References [Enhancements On Off] (What's this?)

  • [DH] S. Diaz and J. Harris, Geometry of Severi Varieties, Trans. Amer. Math. Soc. 309, No. 1 (1988), 1-34.
  • [GH] P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Willey & Sons, 1994.
  • [I] E. Ionel, Genus-One Enumerative Invariants in $\mathbb{P}^n$ with Fixed $j$-Invariant, Duke Math. J. 94 (1998), no. 2, 279-324.
  • [KM] M. Kontsevich and Yu. Manin, Gromov-Witten Classes, Quantum Cohomology, and Enumerative Geometry, Comm. Math. Phys. 164 (1994), no. 3, 525-562.
  • [KQR] S. Katz, Z. Qin, and Y. Ruan, Enumeration of Nodal Genus-$2$ Plane Curves with Fixed Complex Structure, J. Algebraic Geom. 7 (1998), no. 3, 569-587.
  • [P] R. Pandharipande, Counting Elliptic Plane Curves with Fixed $j$-Invariant, Proc. Amer. Math. Soc. 125 (1997), no. 12, 3471-3479.
  • [RT] Y. Ruan and G. Tian, A Mathematical Theory of Quantum Cohomology, J. Diff. Geom. 42 (1995), no. 2, 259-367.
  • [V1] R. Vakil, Enumerative Geometry of Plane Curve of Low Genus, AG/9803007.
  • [V2] R. Vakil, A Tool for Stable Reduction of Curves on Surfaces, Advances in Algebraic Geometry Motivated by Physics, 145-154, Amer. Math. Soc., 2001.
  • [Z] A. Zinger, Enumeration of Genus-Two Curves with a Fixed Complex Structure in $\mathbb{P}^2$ and $\mathbb{P}^3$, math.SG/0201254.

Additional Information

Aleksey Zinger
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Rm 2-586, Cambridge, Massachusetts 02139
Address at time of publication: Department of Mathematics, Stanford University, Stanford, California 94305-2125

Received by editor(s): February 1, 2002
Published electronically: December 8, 2003
Additional Notes: Partially supported by an NSF Graduate Research Fellowship and NSF grant DMS-9803166