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Counting tropical elliptic plane curves with fixed \(j\)-invariant. (English) Zbl 1205.14071
R. Pandharipande [Proc. Am. Math. Soc. 125, No.12, 3471–3479 (1997; Zbl 0883.14031)] counted algebraic elliptic curves of given degree, with given \(j\)-invariant, and passing through given generic points in the plane – the answer is independent of the value of \(j\)-invariant, unless the latter equals 0 or 1728. The authors prove that this answer coincides with the number of tropical elliptic curves of given degree, with given \(j\)-invariant, and passing through given generic points (the tropical curves should be counted with natural multiplicities, coming from the moduli space of tropical elliptic curves, constructed in the paper). In particular, this weighted number of tropical curves is independent of the value of \(j\)-invariant and position of points. This gives a generalization of G. Mikhalkin’s tropical enumeration [J. Am. Math. Soc. 18, 313–377 (2005; Zbl 1092.14068)] to elliptic curves with given \(j\)-invariant (although no explicit correspondence between tropical and classical elliptic curves is established). Immediate applications include a simplification of Mikhalkin’s lattice path count for curves in the projective plane.

MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14T05 Tropical geometry (MSC2010)
51M20 Polyhedra and polytopes; regular figures, division of spaces
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
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