Let K be an algebraically closed, complete non-Archimedean field. The purpose of this paper is to carefully study the extent to which finite morphisms of algebraic K-curves are controlled by certain combinatorial objects, called skeleta. A skeleton is a metric graph embedded in the Berkovich analytification of X. A skeleton has the natural structure of a metrized complex of curves. We prove that a finite morphism of K-curves gives rise to a finite harmonic morphism of a suitable choice of skeleta. We use this to give analytic proofs of stronger 'skeletonized' versions of some foundational results of Liu-Lorenzini, Coleman, and Liu on simultaneous semistable reduction of curves. We then consider the inverse problem of lifting finite harmonic morphisms of metrized complexes to morphisms of curves over K. We prove that every tamely ramified finite harmonic morphism of -metrized complexes of k-curves lifts to a finite morphism of K-curves. 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. As an application, we discuss the relationship between harmonic morphisms of metric graphs and induced maps between component groups of Néron models, providing a negative answer to a question of Ribet motivated by number theory. This article is the first in a series of two. The second article contains several applications of our lifting results to questions about lifting morphisms of tropical curves.
-We give a tropical interpretation of Hurwitz numbers extending the one discovered in [CJM]. In addition we treat a generalization of Hurwitz numbers for surfaces with boundary which we call open Hurwitz numbers.Hurwitz numbers are defined as the (weighted) number of ramified coverings of a compact closed oriented surface S of a given genus having a given set of critical values with given ramification profiles. These numbers have a long history, and have connections to many areas of mathematics, among which we can mention algebraic geometry, topology, combinatorics, and representation theory (see [LZ04] for example).Here we define a slight generalization of these numbers that we call open Hurwitz numbers. To do so, we fix not only points on S and ramification profiles, but also a collection of disjoint circles on S and the behavior of the coverings above each of these circles. Note that the total space of the ramified coverings we consider now is allowed to have boundary components.We also define tropical open Hurwitz numbers, and establish a correspondence with their complex counterpart. This can simply be seen as a
ABSTRACT. In this paper we prove several lifting theorems for morphisms of tropical curves. We interpret the obstruction to lifting a finite harmonic morphism of augmented metric graphs to a morphism of algebraic curves as the non-vanishing of certain Hurwitz numbers, and we give various conditions under which this obstruction does vanish. In particular we show that any finite harmonic morphism of (nonaugmented) metric graphs lifts. We also give various applications of these results. For example, we show that linear equivalence of divisors on a tropical curve C coincides with the equivalence relation generated by declaring that the fibers of every finite harmonic morphism from C to the tropical projective line are equivalent. We study liftability of metrized complexes equipped with a finite group action, and use this to classify all augmented metric graphs arising as the tropicalization of a hyperelliptic curve. We prove that there exists a d-gonal tropical curve that does not lift to a d-gonal algebraic curve.This article is the second in a series of two.Throughout this paper, unless explicitly stated otherwise, K denotes a complete algebraically closed nonArchimedean field with nontrivial valuation val : K → R ∪ {∞}. Its valuation ring is denoted R, its maximal ideal is mR, and the residue field is k = R/mR. We denote the value group of K by Λ = val(K × ) ⊂ R.
Abstract. Welschinger invariants of the real projective plane can be computed via the enumeration of enriched graphs, called marked floor diagrams. By a purely combinatorial study of these objects, we establish a Caporaso-Harris type formula which allows one to compute Welschinger invariants for configurations of points with any number of complex conjugated points.
We prove the vanishing of many Welschinger invariants of real symplectic 4-manifolds.In some particular instances, we also determine their sign and show that they are divisible by a large power of 2. Those results are a consequence of several relations among Welschinger invariants obtained by a real version of symplectic sum formula. In particular, this note contains proofs of results announced in [BP13].A real symplectic manifold (X, ω, τ ) is a symplectic manifold (X, ω) equipped with an antisymplectic involution τ . The real part of (X, ω, τ ), denoted by RX, is by definition the fixed point set of τ . We say that an almost complex structure J tamed byLet X R = (X, ω, τ ) be a real symplectic manifold of dimension 4. Let C be an immersed real rational J-holomorphic curve in X for some τ -compatible almost complex structure J, and denote by L the connected component of RX containing the 1-dimensional part RC of RC. Fix also a τ -invariant class F in H 2 (X \ L; Z/2Z). Any half of C \ RC defines a class C in H 2 (X, L; Z/2Z) whose intersection number modulo 2 with F , denoted by C · F , is well defined and does not depend on the chosen half. We further denote by m(C) the number of nodes of C in L with two τ -conjugated branches, and we define the F -mass of C asChoose a connected component L of RX, a class d ∈ H 2 (X; Z), and r, s ∈ Z ≥0 such that c 1 (X) · d − 1 = r + 2s.Choose a configuration x made of r points in L and s pairs of τ -conjugated points in X \ RX. Given a τ -compatible almost complex structure J, we denote by C(d, x, J) the set of real rational J-holomorphic curves in X realizing the class d, passing through x, and such that L contains RC. For a generic choice of J, the set C(d, x, J) is finite, and the integerdepends neither on x, J, nor on the deformation class of X R (see [Wel05, IKS13]) 1 . We call these numbers the Welschinger invariants of X R . When F = [RX \ L], we simply denote W X R ,L (d, s) instead of W X R ,L,[RX\L] (d, s). Note that Welschinger invariants are non-trivial to compute only in the case of rational manifolds.
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