Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Enumeration of genus-three plane curves with a fixed complex structure

Author: Aleksey Zinger
Journal: J. Algebraic Geom. 14 (2005), 35-81
Published electronically: May 24, 2004
MathSciNet review: 2092126
Full-text PDF

Abstract | References | Additional Information

Abstract: We give a practical formula for counting irreducible nodal genus-three plane curves with a fixed general complex structure on the normalization. As an intermediate step, we enumerate rational plane curves that have a $(3,4)$-cusp.

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

  • [AF1] P. Aluffi and C. Faber, Linear Orbits of Smooth Plane Curves, J. Algebraic Geom. 2 (1993), no. 1, 155-184. MR 1185610 (94e:14032)
  • [AF2] -, Linear Orbits of Arbitrary Plane Curves, Mich. Math. J. 48 (2000), 1-37. MR 1786478 (2001h:14068)
  • [ACGH] E. Arbarello, M. Cornalba, P. Griffiths, J. Harris, Geometry of Algebraic Curves, Vol. I, Springer-Verlag, New York, 1985. MR 0770932 (86h:14019)
  • [FO] K. Fukaya and K. Ono, Arnold Conjecture and Gromov-Witten Invariant, Topology 38 (1999), no. 5, 933-1048. MR 1688434 (2000j:53116)
  • [GH] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Library Edition, 1994. MR 1288523 (95d:14001)
  • [I] E. Ionel, Genus-One Enumerative Invariants in $\mathbb{P}^n$ with Fixed $j$-Invariant, Duke Math. J. 94 (1998), no. 2, 279-324. MR 1638587 (2000h:53113)
  • [KM] M. Kontsevich and Yu. Manin, Gromov-Witten Classes, Quantum Cohomology, and Enumerative Geometry, Comm. Math. Phys. 164 (1994), no. 3, 525-562. MR 1291244 (95i:14049)
  • [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. MR 1618136 (99d:14058)
  • [LT] J. Li and G. Tian, Virtual Moduli Cycles and Gromov-Witten Invariants of General Symplectic Manifolds, Topics in Symplectic $4$-manifolds, 47-83. MR 1635695 (2000d:53137)
  • [P1] R. Pandharipande, Counting Elliptic Plane Curves with Fixed $j$-Invariant, Proc. Amer. Math. Soc. 125 (1997), no. 12, 3471-3479. MR 1423328 (98h:14068)
  • [P2] -, Intersections of ${\mathbf{Q}}$-Divisors on Kontsevich's Moduli Space $\bar{M}_{0,n}(P^r,d)$ and Enumerative Geometry, Trans. Amer. Math. Soc. 351 (1999), no. 4, 1481-1505. MR 1407707 (99f:14068)
  • [RT] Y. Ruan and G. Tian, A Mathematical Theory of Quantum Cohomology, J. Diff. Geom. 42 (1995), no. 2, 259-367. MR 1366548 (96m:58033)
  • [Z1] A. Zinger, Completion of Katz-Qin-Ruan's Enumeration of Genus-Two Plane Curves, J. Alg. Geom. 13, no. 3, pp. 547-561.
  • [Z2] -, Enumeration of Genus-Two Curves with a Fixed Complex Structure in $\mathbb{P}^2$ and $\mathbb{P}^3$, math.SG/0201254.
  • [Z3] -, Enumerative vs. Symplectic Invariants and Obstruction Bundles, math.SG/0201255.

Additional Information

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

Received by editor(s): September 7, 2002
Received by editor(s) in revised form: December 17, 2002
Published electronically: May 24, 2004
Additional Notes: Partially supported by NSF grant DMS-9803166

American Mathematical Society