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Lifting harmonic morphisms of tropical curves, metrized complexes, and Berkovich skeleta
, 2013
"... Let K be a complete and algebraically closed field with value group Λ and residue field k, and let ϕ: X ′ → X be a finite morphism of smooth, proper, irreducible, stable marked algebraic curves over K. We show that ϕ gives rise in a canonical way to a finite and effective harmonic morphism of Λm ..."
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Let K be a complete and algebraically closed field with value group Λ and residue field k, and let ϕ: X ′ → X be a finite morphism of smooth, proper, irreducible, stable marked algebraic curves over K. We show that ϕ gives rise in a canonical way to a finite and effective harmonic morphism of Λmetric graphs, and more generally to a finite harmonic morphism of Λmetrized complexes of kcurves. These canonical “abstract tropicalizations ” are constructed using Berkovich’s notion of the skeleton of an analytic curve. Our arguments give analytic proofs of stronger “skeletonized ” versions of some foundational results of LiuLorenzini, Coleman, and Liu on simultaneous semistable reduction of curves. We then consider the inverse problem of lifting finite harmonic morphisms of metric graphs/tropical curves and metrized complexes to morphisms of curves over K. We prove that every tamely ramified finite harmonic morphism of Λmetrized complexes of kcurves lifts to a finite morphism of Kcurves. If in addition the ramification points are marked, we obtain a complete classification of all such lifts along with their automorphisms. This generalizes and provides new analytic proofs of earlier results of Saïdi and Wewers. We prove a similar result concerning the existence of liftings for morphisms of tropical curves, except the genus of the source curve can no longer be fixed. From this point of view, morphisms of metrized complexes are better behaved than morphisms of tropical curves. The caveat on the genus in the lifting
Tropical mirror symmetry for elliptic curves. arXiv preprint arXiv:1309.5893
, 2013
"... Abstract. Mirror symmetry relates GromovWitten invariants of an elliptic curve with certain integrals over Feynman graphs [9]. We prove a tropical generalization of mirror symmetry for elliptic curves, i.e., a statement relating certain labeled GromovWitten invariants of a tropical elliptic curve ..."
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Abstract. Mirror symmetry relates GromovWitten invariants of an elliptic curve with certain integrals over Feynman graphs [9]. We prove a tropical generalization of mirror symmetry for elliptic curves, i.e., a statement relating certain labeled GromovWitten invariants of a tropical elliptic curve to more refined Feynman integrals. This result easily implies the tropical analogue of the mirror symmetry statement mentioned above and, using the necessary Correspondence Theorem, also the mirror symmetry statement itself. In this way, our tropical generalization leads to an alternative proof of mirror symmetry for elliptic curves. We believe that our approach via tropical mirror symmetry naturally carries the potential of being generalized to more adventurous situations of mirror symmetry. Moreover, our tropical approach has the advantage that all involved invariants are easy to compute. Furthermore, we can use the techniques for computing Feynman integrals to prove that they are quasimodular forms. Also, as a side product, we can give a combinatorial characterization of Feynman graphs for which the corresponding integrals are zero. More generally, the tropical mirror symmetry theorem gives a natural interpretation of the Amodel side (i.e., the generating function of GromovWitten invariants) in terms of a sum over Feynman graphs. Hence our quasimodularity result becomes meaningful on the Amodel side as well. Our theoretical results are complemented by a Singular package including several procedures that can be used to compute Hurwitz numbers of the elliptic curve as integrals over Feynman graphs. Contents
Tropical covers of curves and their moduli spaces
 Comm. Contemp. Math
, 2013
"... Abstract. We define the tropical moduli space of covers of a tropical line in the plane as weighted abstract polyhedral complex, and the tropical branch map recording the images of the simple ramifications. Our main result is the invariance of the degree of the branch map, which enables us to give a ..."
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Abstract. We define the tropical moduli space of covers of a tropical line in the plane as weighted abstract polyhedral complex, and the tropical branch map recording the images of the simple ramifications. Our main result is the invariance of the degree of the branch map, which enables us to give a tropical intersectiontheoretic definition of tropical triple Hurwitz numbers. We show that our intersectiontheoretic definition coincides with the one given in [3] where a Correspondence Theorem for Hurwitz numbers is proved. Thus we provide a tropical intersectiontheoretic justification for the multiplicities with which a tropical cover has to be counted. Our method of proof is to establish a local duality between our tropical moduli spaces and certain moduli spaces of relative stable maps to P1. 1.
THREE TROPICAL ENUMERATIVE PROBLEMS
"... Abstract. In this survey, we describe three tropical enumerative problems and the corresponding moduli spaces of tropical curves. They have the structure of weighted polyhedral complexes. We observe similarities in the definitions of the weights, aiming at a better understanding of the tropical str ..."
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Abstract. In this survey, we describe three tropical enumerative problems and the corresponding moduli spaces of tropical curves. They have the structure of weighted polyhedral complexes. We observe similarities in the definitions of the weights, aiming at a better understanding of the tropical structure of the moduli spaces. 1. introduction In tropical geometry, algebraic varieties are degenerated to certain piecewise linear objects called tropical varieties. This process loses a lot of information, but many properties of the algebraic variety can be read off the tropical variety, and many theorems that hold for the algebraic side remarkably continue to hold on the tropical side. Since tropical varieties are piecewise linear, they are in principle easier to understand than algebraic varieties and combinatorial methods apply. Thus there is hope that we can use tropical geometry to derive theorems in algebraic geometry. One of the fields in which tropical geometry has had significant success
THE DOUBLE GROMOVWITTEN INVARIANTS OF HIRZEBRUCH SURFACES ARE PIECEWISE POLYNOMIAL
"... Abstract. We define the double GromovWitten invariants of Hirzebruch surfaces in analogy with double Hurwitz numbers, and we prove that they satisfy a piecewise polynomiality property analogous to their 1dimensional counterpart. Furthermore we show that each polynomial piece is either even or odd ..."
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Abstract. We define the double GromovWitten invariants of Hirzebruch surfaces in analogy with double Hurwitz numbers, and we prove that they satisfy a piecewise polynomiality property analogous to their 1dimensional counterpart. Furthermore we show that each polynomial piece is either even or odd, and we compute its degree. Our methods combine floor diagrams and Ehrhart theory. 1.
pADIC HURWITZ NUMBERS
, 806
"... Abstract. We introduce stable tropical curves, and use these to count covers of the padic projective line of fixed degree and ramification types by Mumford curves in terms of tropical Hurwitz numbers. Our counts depend on the branch loci of the covers. 1. ..."
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Abstract. We introduce stable tropical curves, and use these to count covers of the padic projective line of fixed degree and ramification types by Mumford curves in terms of tropical Hurwitz numbers. Our counts depend on the branch loci of the covers. 1.
Bijections for simple and double Hurwitz numbers
"... Abstract. We give a bijective proof of Hurwitz formula for the number of simple branched coverings of the sphere by itself. Our approach extends to double Hurwitz numbers and yields new properties for them. In particular we prove for double Hurwitz numbers a conjecture of Kazarian and Zvonkine, and ..."
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Abstract. We give a bijective proof of Hurwitz formula for the number of simple branched coverings of the sphere by itself. Our approach extends to double Hurwitz numbers and yields new properties for them. In particular we prove for double Hurwitz numbers a conjecture of Kazarian and Zvonkine, and we give an expression that in a sense interpolates between two celebrated polynomiality properties: polynomiality in chambers for double Hurwitz numbers, and a new analog for almost simple genus 0 Hurwitz numbers of the polynomiality up to normalization of simple Hurwitz numbers of genus g. Some probabilistic implications of our results for random branched coverings are briefly discussed in conclusion. 1