Enumeration of genus-three plane curves with a fixed complex structure
Author:
Aleksey Zinger
Journal:
J. Algebraic Geom. 14 (2005), 35-81
DOI:
https://doi.org/10.1090/S1056-3911-04-00375-3
Published electronically:
May 24, 2004
MathSciNet review:
2092126
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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.
- Paolo Aluffi and Carel Faber, Linear orbits of smooth plane curves, J. Algebraic Geom. 2 (1993), no. 1, 155–184. MR 1185610
- Paolo Aluffi and Carel Faber, Linear orbits of arbitrary plane curves, Michigan Math. J. 48 (2000), 1–37. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786478, DOI https://doi.org/10.1307/mmj/1030132706
- E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932
- Kenji Fukaya and Kaoru Ono, Arnold conjecture and Gromov-Witten invariant, Topology 38 (1999), no. 5, 933–1048. MR 1688434, DOI https://doi.org/10.1016/S0040-9383%2898%2900042-1
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. MR 1288523
- Eleny Ionel, Genus $1$ enumerative invariants in $\mathbf P^n$ with fixed $j$ invariant, Duke Math. J. 94 (1998), no. 2, 279–324. MR 1638587, DOI https://doi.org/10.1215/S0012-7094-98-09414-5
- M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562. MR 1291244
- Sheldon Katz, Zhenbo Qin, and Yongbin Ruan, Enumeration of nodal genus-$2$ plane curves with fixed complex structure, J. Algebraic Geom. 7 (1998), no. 3, 569–587. MR 1618136
- Jun Li and Gang Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, Topics in symplectic $4$-manifolds (Irvine, CA, 1996) First Int. Press Lect. Ser., I, Int. Press, Cambridge, MA, 1998, pp. 47–83. MR 1635695
- Rahul Pandharipande, Counting elliptic plane curves with fixed $j$-invariant, Proc. Amer. Math. Soc. 125 (1997), no. 12, 3471–3479. MR 1423328, DOI https://doi.org/10.1090/S0002-9939-97-04136-1
- Rahul Pandharipande, Intersections of $\mathbf Q$-divisors on Kontsevich’s moduli space $\overline M_{0,n}(\mathbf P^r,d)$ and enumerative geometry, Trans. Amer. Math. Soc. 351 (1999), no. 4, 1481–1505. MR 1407707, DOI https://doi.org/10.1090/S0002-9947-99-01909-1
- Yongbin Ruan and Gang Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), no. 2, 259–367. MR 1366548
[Z1]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]Z2 ---, Enumeration of Genus-Two Curves with a Fixed Complex Structure in $\mathbb {P}^2$ and $\mathbb {P}^3$, math.SG/0201254.
[Z3]Z3 ---, Enumerative vs. Symplectic Invariants and Obstruction Bundles, math.SG/0201255.
[AF1]AF P. Aluffi and C. Faber, Linear Orbits of Smooth Plane Curves, J. Algebraic Geom. 2 (1993), no. 1, 155–184.
[AF2]AF2 ---, Linear Orbits of Arbitrary Plane Curves, Mich. Math. J. 48 (2000), 1–37.
[ACGH]ACGH E. Arbarello, M. Cornalba, P. Griffiths, J. Harris, Geometry of Algebraic Curves, Vol. I, Springer-Verlag, New York, 1985.
[FO]FO K. Fukaya and K. Ono, Arnold Conjecture and Gromov-Witten Invariant, Topology 38 (1999), no. 5, 933–1048.
[GH]GH P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Library Edition, 1994.
[I]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]KM M. Kontsevich and Yu. Manin, Gromov-Witten Classes, Quantum Cohomology, and Enumerative Geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562.
[KQR]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.
[LT]LT J. Li and G. Tian, Virtual Moduli Cycles and Gromov-Witten Invariants of General Symplectic Manifolds, Topics in Symplectic $4$-manifolds, 47–83.
[P1]P1 R. Pandharipande, Counting Elliptic Plane Curves with Fixed $j$-Invariant, Proc. Amer. Math. Soc. 125 (1997), no. 12, 3471–3479.
[P2]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.
[RT]RT Y. Ruan and G. Tian, A Mathematical Theory of Quantum Cohomology, J. Diff. Geom. 42 (1995), no. 2, 259–367.
[Z1]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]Z2 ---, Enumeration of Genus-Two Curves with a Fixed Complex Structure in $\mathbb {P}^2$ and $\mathbb {P}^3$, math.SG/0201254.
[Z3]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
Email:
azinger@math.stanford.edu
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
