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Skeletons of stable maps. I: rational curves in toric varieties. (English) Zbl 1401.14130
The tight connections between moduli spaces in algebraic and tropical geometry have received lots of attention in recent years. This paper unites and extends all the existing methods in this area, attaining a relation between moduli spaces of log-stable rational maps to toric varieties and their tropical counterparts, which builds on toroidal geometry and non-Archimedean analytic geometry. The result has several interesting consequences and offers new perspectives both for algebraic and tropical geometry. For example, the algebraic moduli space of log-stable rational maps to a toric variety can be viewed as a tropical compactification, or toroidal modification. In tropical geometry, correspondence theorems – which are at the heart of any successful application of tropical methods in enumerative geometry – can be deduced from the relation between these moduli spaces.
For part II, see [D. Ranganathan, Res. Math. Sci. 4, Paper No. 11, 18 p. (2017; Zbl 1401.14131)].

14G22 Rigid analytic geometry
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14T05 Tropical geometry (MSC2010)
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