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Enumerative geometry of elliptic curves on toric surfaces

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Len_2018_Enumerative_geometry_of_IJM_AAM.pdf (552.1Kb)
Date
06/2018
Author
Len, Y.
Ranganathan, D.
Keywords
QA Mathematics
T-NDAS
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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.
Citation
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
Publication
Israel Journal of Mathematics
Status
Peer reviewed
DOI
https://doi.org/10.1007/s11856-018-1698-9
ISSN
0021-2172
Type
Journal article
Rights
Copyright © 2018, The Hebrew University of Jerusalem. This work has been made available online in accordance with the publisher’s policies. This is the author created accepted version manuscript following peer review and as such may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1007/s11856-018-1698-9
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  • University of St Andrews Research
URL
https://arxiv.org/abs/1510.08556
URI
http://hdl.handle.net/10023/20196

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