Counting tropical elliptic plane curves with fixed j-invariant

Michael Kerber, Hannah Markwig
2009 Commentarii Mathematici Helvetici  
In complex algebraic geometry, the problem of enumerating plane elliptic curves of given degree with fixed complex structure has been solved by R. Pandharipande [8] using Gromov-Witten theory. In this article we treat the tropical analogue of this problem, the determination of the number E trop .d / of tropical elliptic plane curves of degree d and fixed "tropical j -invariant" interpolating an appropriate number of points in general position and counted with multiplicities. We show that this
more » ... mber is independent of the position of the points and the value of the j -invariant and that it coincides with the number of complex elliptic curves (with j -invariant j ... f0; 1728g). The result can be used to simplify G. Mikhalkin's algorithm to count curves via lattice paths (see [6] ) in the case of rational plane curves. in the cases where the complex j -invariant is j 2 f0; 1728g. The methods of our computation of E trop .d / using a very large j -invariant are analogous to Pandharipande's computation of the numbers E.d; j / -we use moduli spaces of tropical elliptic curves and evaluation maps. But we can also compute E trop .d / as mentioned above in another way, using a very small j -invariant. Tropical curves with a very small j -invariant can be related to rational curves, too. Thus we can determine the number E trop .d / with the aid of G. Mikhalkin's lattice path count (see Theorem 2 of [6]). The computation of E trop .d / using the very small j -invariant does not have a counterpart in complex algebraic geometry. Vol. 84 (2009) Counting tropical elliptic plane curves with fixed j -invariant Example 2.2. We want to determine the space M trop; 1;1 . An element of M trop; 1;1 is an abstract tropical curve with one unbounded edge, and of genus 1. As no divalent vertices are allowed, such an abstract tropical curve consists of one bounded edge whose two endpoints are identified and glued to the unbounded edge. These curves only differ in the length of their bounded edge, which has to be positive. Therefore M trop; 1;1 is isomorphic to the open interval .0; 1/. We define x M trop; 1;1 to be the interval OE0; 1/. Following G. Mikhalkin, we call the length of the bounded edge -which is an inner invariant of the tropical elliptic curve -its tropical j -invariant, as it plays the role of the j -invariant of elliptic curves in algebraic geometry (see Example 3.15 of [7], see also Remark 2.6 and Definition 4.2). Definition 2.3. An n-marked plane tropical curve of genus g is a tuple .; h; x 1 ; : : : ; x n /; Vol. 84 (2009) Counting tropical elliptic plane curves with fixed j -invariant
doi:10.4171/cmh/166 fatcat:buzinpsofzfyvlgp7b74rnemvy