Skip to content

Research at St Andrews

Enumerative geometry of elliptic curves on toric surfaces

Research output: Contribution to journalArticlepeer-review

Open Access permissions

Open

Author(s)

Y. Len, D. Ranganathan

School/Research organisations

Abstract

We establish the equality of classical and tropical curve counts for elliptic curves on toric surfaces with fixed j-invariant, refining results of Mikhalkin and Nishinou--Siebert. As an application, we determine a formula for such counts on ℙ2 and all Hirzebruch surfaces. This formula relates the count of elliptic curves with the number of rational curves on the surface satisfying a small number of tangency conditions with the toric boundary. Furthermore, the combinatorial tropical multiplicities of Kerber and Markwig for counts in ℙ2 are derived and explained algebro-geometrically, using Berkovich geometry and logarithmic Gromov--Witten theory. As a consequence, a new proof of Pandharipande's formula for counts of elliptic curves in ℙ2 with fixed j-invariant is obtained.
Close

Details

Original languageEnglish
Pages (from-to)351–385
JournalIsrael Journal of Mathematics
Volume226
Early online date11 May 2018
DOIs
Publication statusPublished - Jun 2018

Discover related content
Find related publications, people, projects and more using interactive charts.

View graph of relations

Related by author

  1. Prym-Brill-Noether loci of special curves

    Creech, S., Len, Y., Ritter, C. & Wu, D., 25 Aug 2020, In: International Mathematics Research Notices. Advance Articles, 41 p.

    Research output: Contribution to journalArticlepeer-review

  2. Lifting tropical self intersections

    Len, Y. & Satriano, M., Feb 2020, In: Journal of Combinatorial Theory, Series A. 170, 21 p., 105138.

    Research output: Contribution to journalArticlepeer-review

  3. Lifting tropical bitangents

    Len, Y. & Markwig, H., Jan 2020, In: Journal of Symbolic Computation. 96, p. 122-152

    Research output: Contribution to journalArticlepeer-review

  4. Projective duals to algebraic and tropical hypersurfaces

    Ilten, N. & Len, Y., Nov 2019, In: Proceedings of the London Mathematical Society. 119, 5, p. 1234-1278

    Research output: Contribution to journalArticlepeer-review

  5. Bitangents of non-smooth tropical quartics

    Lee, H. & Len, Y., 5 Jul 2018, In: Portugaliae Mathematica. 75, 1, p. 67-78

    Research output: Contribution to journalArticlepeer-review

Related by journal

  1. Integrals of groups

    Araújo, J., Cameron, P. J., Casolo, C. & Matucci, F., Oct 2019, In: Israel Journal of Mathematics. 234, 1, p. 149-178

    Research output: Contribution to journalArticlepeer-review

  2. The pressure function for infinite equilibrium measures

    Bruin, H., Terhesiu, D. & Todd, M., Aug 2019, In: Israel Journal of Mathematics. 232, 2, p. 775-826 52 p.

    Research output: Contribution to journalArticlepeer-review

  3. Distance sets, orthogonal projections, and passing to weak tangents

    Fraser, J. M., Jun 2018, In: Israel Journal of Mathematics. 226, 2, p. 851–875 25 p.

    Research output: Contribution to journalArticlepeer-review

  4. Chains of subsemigroups

    Cameron, P. J., Gadouleau, M., Mitchell, J. D. & Peresse, Y., Jun 2017, In: Israel Journal of Mathematics. 220, 1, p. 479-508

    Research output: Contribution to journalArticlepeer-review

ID: 268424557

Top