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Enumerative geometry of elliptic curves on toric surfaces

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Enumerative geometry of elliptic curves on toric surfaces. / Len, Y.; Ranganathan, D.

In: Israel Journal of Mathematics, Vol. 226, 06.2018, p. 351–385.

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Len, Y & Ranganathan, D 2018, 'Enumerative geometry of elliptic curves on toric surfaces', Israel Journal of Mathematics, vol. 226, pp. 351–385. https://doi.org/10.1007/s11856-018-1698-9

APA

Len, Y., & Ranganathan, D. (2018). Enumerative geometry of elliptic curves on toric surfaces. Israel Journal of Mathematics, 226, 351–385. https://doi.org/10.1007/s11856-018-1698-9

Vancouver

Len Y, Ranganathan D. Enumerative geometry of elliptic curves on toric surfaces. Israel Journal of Mathematics. 2018 Jun;226:351–385. https://doi.org/10.1007/s11856-018-1698-9

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Len, Y. ; Ranganathan, D. / Enumerative geometry of elliptic curves on toric surfaces. In: Israel Journal of Mathematics. 2018 ; Vol. 226. pp. 351–385.

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@article{f252a61fbe254d77ac87bf7fb1d76ce5,
title = "Enumerative geometry of elliptic curves on toric surfaces",
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.",
author = "Y. Len and D. Ranganathan",
year = "2018",
month = jun,
doi = "10.1007/s11856-018-1698-9",
language = "English",
volume = "226",
pages = "351–385",
journal = "Israel Journal of Mathematics",
issn = "0021-2172",
publisher = "HEBREW UNIV MAGNES PRESS",

}

RIS (suitable for import to EndNote) - Download

TY - JOUR

T1 - Enumerative geometry of elliptic curves on toric surfaces

AU - Len, Y.

AU - Ranganathan, D.

PY - 2018/6

Y1 - 2018/6

N2 - 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.

AB - 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.

U2 - 10.1007/s11856-018-1698-9

DO - 10.1007/s11856-018-1698-9

M3 - Article

VL - 226

SP - 351

EP - 385

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

ER -

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