## Arkiv för Matematik

• Ark. Mat.
• Volume 39, Number 1 (2001), 157-180.

### Recursions for characteristic numbers of genus one plane curves

Ravi Vakil

#### Abstract

Characteristic numbers of families of maps of nodal curves to P2 are defined as intersection of natural divisor classes. (This definition agrees with the usual definition for families of plane curves.) Simple recursions for characteristic numbers of genus one plane curves of all degrees are computed.

#### Article information

Source
Ark. Mat., Volume 39, Number 1 (2001), 157-180.

Dates
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485898715

Digital Object Identifier
doi:10.1007/BF02388797

Mathematical Reviews number (MathSciNet)
MR1821088

Zentralblatt MATH identifier
1069.14059

Rights

#### Citation

Vakil, Ravi. Recursions for characteristic numbers of genus one plane curves. Ark. Mat. 39 (2001), no. 1, 157--180. doi:10.1007/BF02388797. https://projecteuclid.org/euclid.afm/1485898715

#### References

• [A1] Aluffi, P., The characteristic numbers of smooth plane cubics, in Algebraic Geometry (Sundance, 1986) (Holme, A. and Speiser, R., eds.), Lecture Notes in Math. 1311, pp. 1–8, Springer-Verlag, Berlin-New York, 1988.
• [A2] Aluffi, P., How many smooth plane cubics with given j-invariant are tangent to 8 lines in general position?, in Enumerative Algebraic Geometry (Copenhagen, 1989) (Kleiman, S. L. and Thorup, A., eds.), Contemp. Math. 123, pp. 15–29, Amer. Math. Soc., Providence, R. I., 1991.
• [A3] Aluffi, P., Some characteristic numbers for nodal and cuspidal plane curves of any degree, Manuscripta Math. 72 (1991), 425–444.
• [CH] Caporaso, L. and Harris, J., Counting plane curves of any genus, Invent. Math. 131 (1998), 345–392.
• [CT] Crescimanno, M. and Taylor, W., Large N phases of chiral QCD2, Nuclear Phys. B 437 (1995), 3–24.
• [DM] Deligne, P. and Mumford, D., The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–110.
• [DH] Diaz, S. and Harris, J., Geometry of the Severi variety, Trans. Amer. Math. Soc. 309 (1988), 1–34.
• [DZ] Dubrovin, B. and Zhang, Y., Bi-Hamiltonian hierarchies in 2D topological field theory at one-loop approximation. Comm. Math. Phys. 198 (1998), 311–361.
• [EK1] Ernström, L. and Kennedy, G., Recursive formulas for the characteristic numbers of rational plane curves, J. Algebraic Geom. 7 (1998), 141–181.
• [EK2] Ernström, L. and Kennedy, G., Contact cohomology of the projective plane, Amer. J. Math. 121 (1999), 73–96.
• [F] Fulton, W., Intersection Theory, Springer-Verlag, New York, 1984.
• [FP] Fulton, W. and Pandharipande, R., Notes on stable maps and quantum cohomology, in Algebraic Geometry—Santa Cruz 1995 (Kollár, J., Lazarsfeld, R. and Morrison, D. R., eds.), Proc. Symp. Pure Math. 62, Part 2, pp. 45–96, Amer. Math. Soc., Providence, R. I., 1997.
• [GJ1] Goulden, I. P. and Jackson, D. M., Transitive factorisations into transpositions and holomorphic mappings on the sphere, Proc. Amer. Math. Soc. 125 (1997), 51–60.
• [GJ2] Goulden, I. P. and Jackson, D. M., The number of ramified coverings of the sphere by the double torus, and a general form for higher genera, J. Combin. Theory Ser. A 88 (1999), 259–275.
• [GJV] Goulden, I. P., Jackson, D. M. and Vakil, R., The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals, to appear in Proc. London Math. Soc.
• [GP] Graber, T. and Pandharipande, R., Personal communication, 1997.
• [GD] Grothendieck, A. and Dieudonné, J., Eléments de géometrie algébrique, Inst. Hautes Études Sci. Publ. Math. 4, 8, 11, 17, 20, 24, 28, 32 (1961–1967).
• [H] Harris, J., On the Severi problem, Invent. Math. 84 (1986), 445–461.
• [HM] Harris, J. and Morrison, I., Moduli of Curves. Springer-Verlag, New York, 1998.
• [Ha] Hartshorne, R., Algebraic Geometry, Springer-Verlag, New York, 1977.
• [J] de Jong, A. J., Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93.
• [KK] Kabanov, A. and Kimura, T., Intersection numbers and rank one cohomological field theories in genus one, Comm. Math. Phys. 194 (1998), 651–674.
• [K1] Kleiman, S. L., Problem 15: Rigorous foundation of Schubert's enumerative calculus, in Mathematical Developments Arising from Hilbert Problems (Northern Ill. Univ., De Kalb, 1974) (Browder, F. E., ed.), Proc. Symp. Pure Math. 28, pp. 445–482, Amer. Math. Soc., Providence, R. I., 1976.
• [K2] Kleiman, S. L., About the conormal scheme, in Complete Intersections (Greco, S. and Strano, R., eds.), Lecture Notes in Math. 1092, pp. 161–197, Springer-Verlag, Berlin-Heidelberg, 1984.
• [K3] Kleiman, S. L., Tangency and duality, in Proceedings of the 1984 Vancouver Conference in Algebraic Geometry (Carrell, J., Geramita, A. V. and Russell, P., eds.), CMS Conf. Proc. 6, pp. 163–225, Amer. Math. Soc., Providence, R. I., 1986.
• [Kn] Knudsen, F. F., The projectivity of the moduli space of stable curves, II: The stacks Mg,n, Math. Scand. 52 (1983), 161–199.
• [KnM] Knudsen, F. F. and Mumford, D., The projectivity of the moduli space of stable curves I, Math. Scand. 39 (1976), 19–55.
• [KM] Kontsevich, M. and Manin, Yu., Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), 525–562.
• [P1] Pandharipande, R., Counting elliptic plane curves with fixed j-invariant, Proc. Amer. Math. Soc. 125 (1997), 3471–3479.
• [P2] Pandharipande, R., The canonical class of $\bar M_{0,n} (\operatorname{P} ^r ,d)$ and enumerative geometry, Internat. Math. Res. Notices 1997, 173–186.
• [P3] Pandharipande, R., Unpublished e-mail to L. Ernstöm, May 2, 1997.
• [P4] Pandharipande, R., Intersection of Q-divisors on Kontsevich’s moduli space $\bar M_{0,n} (\operatorname{P} ^r ,d)$ and enumerative geometry, Trans. Amer. Math. Soc. 351 (1999), 1481–1505.
• [P5] Pandharipande, R., A geometric construction of Getzler's relation, Math. Ann. 313 (1999), 715–729.
• [RT] Ruan, Y. and Tian, G., A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), 295–367.
• [S] Schubert, H., Kalkül der abzählenden Geometrie, Springer-Verlag, Reprint of the 1879 original, Berlin-New York, 1979.
• Thaddeus, M., Personal communication, 1997.
• [V1] Vakil, R., Recursions, formulas, and graph-theoretic interpretations of ramified coverings of the sphere by surfaces of genus 0 and 1, to appear in Trans. Amer. Math. Soc.
• [V2] Vakil, R., The characteristic numbers of quartic plane curves, Canad. J. Math. 51 (1999), 1089–1120.
• [V3] Vakil, R., Counting curves on rational surfaces, to appear in Manuscripta Math. 102 (2000), 53–84.
• [V4] Vakil, R., The enumerative geometry of rational and elliptic plane curves in projective space, J. Reine Angew. Math. 529 (2000), 101–153.
• [V5] Vakil, R., Characteristic numbers of rational and elliptic curves in projective space, In preparation.
• [Vi] Vistoli, A., Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), 613–670.