Lifting problem for minimally wild covers of Berkovich curves

This work continues the study of residually wild morphisms $f\colon Y\to X$ of Berkovich curves initiated by Cohen, Temkin and Trushin in [CTT16]. The different function $\delta_f$ introduced in [CTT16] is the primary discrete invariant of such covers. When $f$ is not residually tame, it provides a non-trivial enhancement of the classical invariant of $f$ consisting of morphisms of reductions $\widetilde{f}\colon \widetilde{Y}\to\widetilde{X}$ and metric skeletons $\Gamma_f\colon \Gamma_Y\to\Gamma_X$. In this paper we interpret $\delta_f$ as the norm of the canonical trace section $\tau_f$ of the dualizing sheaf $\omega_f$, and introduce a finer reduction invariant $\widetilde{\tau}_f$, which is (loosely speaking) a section of $\omega_{\widetilde{f}}^{\rm log}$. Our main result generalizes a lifting theorem of Amini-Baker-Brugall\'e-Rabinoff from the case of residually tame morphism to the case of minimally residually wild morphisms. For such morphisms we describe all restrictions the datum $(\widetilde{f},\Gamma_f,\delta|_{\Gamma_Y},\widetilde{\tau}_f)$ satisfies, and prove that, conversely, any quadruple satisfying these restrictions can be lifted to a morphism of Berkovich curves.Comment: 35 pages, first version, comments are welcom

The tropicalization of the moduli space of curves

We show that the skeleton of the Deligne-Mumford-Knudsen moduli stack of stable curves is naturally identified with the moduli space of extended tropical curves, and that this is compatible with the "naive" set-theoretic tropicalization map. The proof passes through general structure results on the skeleton of a toroidal Deligne-Mumford stack. Furthermore, we construct tautological forgetful, clutching, and gluing maps between moduli spaces of extended tropical curves and show that they are compatible with the analogous tautological maps in the algebraic setting.Comment: v2: 55 pages. Expanded Section 2 with improved treatment of the category of generalized cone complexes. Clarified the role of the coarse moduli space and its analytification in the construction of the skeleton for a toroidal DM stac

Skeletons of stable maps II: Superabundant geometries

We implement new techniques involving Artin fans to study the realizability of tropical stable maps in superabundant combinatorial types. Our approach is to understand the skeleton of a fundamental object in logarithmic Gromov--Witten theory -- the stack of prestable maps to the Artin fan. This is used to examine the structure of the locus of realizable tropical curves and derive 3 principal consequences. First, we prove a realizability theorem for limits of families of tropical stable maps. Second, we extend the sufficiency of Speyer's well-spacedness condition to the case of curves with good reduction. Finally, we demonstrate the existence of liftable genus 1 superabundant tropical curves that violate the well-spacedness condition.Comment: 17 pages, 1 figure. v2 fixes a minor gap in the proof of Theorem D and adds details to the construction of the skeleton of a toroidal Artin stack. Minor clarifications throughout. To appear in Research in the Mathematical Science

Morphisms of Berkovich curves and the different function

Given a generically \'etale morphism $f\colon Y\to X$ of quasi-smooth Berkovich curves, we define a different function $\delta_f\colon Y\to[0,1]$ that measures the wildness of the topological ramification locus of $f$. This provides a new invariant for studying $f$, which cannot be obtained by the usual reduction techniques. We prove that $\delta_f$ is a piecewise monomial function satisfying a balancing condition at type 2 points analogous to the classical Riemann-Hurwitz formula, and show that $\delta_f$ can be used to explicitly construct the simultaneous skeletons of $X$ and $Y$. As an application, we use our results to completely describe the topological ramification locus of $f$ when its degree equals to the residue characteristic $p$.Comment: Final version, 49 pages, to appear in Adv.Mat

Functorial tropicalization of logarithmic schemes: The case of constant coefficients

The purpose of this article is to develop foundational techniques from logarithmic geometry in order to define a functorial tropicalization map for fine and saturated logarithmic schemes in the case of constant coefficients. Our approach crucially uses the theory of fans in the sense of K. Kato and generalizes Thuillier's retraction map onto the non-Archimedean skeleton in the toroidal case. For the convenience of the reader many examples as well as an introductory treatment of the theory of Kato fans are included.Comment: v4: 33 pages. Restructured introduction, otherwise minor changes. To appear in the Proceedings of the LM

Tropical geometry and correspondence theorems via toric stacks

In this paper we generalize correspondence theorems of Mikhalkin and Nishinou-Siebert providing a correspondence between algebraic and parameterized tropical curves. We also give a description of a canonical tropicalization procedure for algebraic curves motivated by Berkovich's construction of skeletons of analytic curves. Under certain assumptions, we construct a one-to-one correspondence between algebraic curves satisfying toric constraints and certain combinatorially defined objects, called "stacky tropical reductions", that can be enumerated in terms of tropical curves satisfying linear constraints. Similarly, we construct a one-to-one correspondence between elliptic curves with fixed $j$-invariant satisfying toric constraints and "stacky tropical reductions" that can be enumerated in terms of tropical elliptic curves with fixed tropical $j$-invariant satisfying linear constraints. Our theorems generalize previously published correspondence theorems in tropical geometry, and our proofs are algebra-geometric. In particular, the theorems hold in large positive characteristic.Comment: Terminology change: "tropical limits" have been changed to "tropical reductions". Minor mistakes have been corrected, and many typos have been fixed. Final version. To appear in Mathematische Annale

Enumerative geometry of elliptic curves on toric surfaces

We establish the equality of classical and tropical curve counts for elliptic curves on toric surfaces with fixed j-invariant, refining results of Mikhalkin and Nishinou--Siebert. As an application, we determine a formula for such counts on ℙ2 and all Hirzebruch surfaces. This formula relates the count of elliptic curves with the number of rational curves on the surface satisfying a small number of tangency conditions with the toric boundary. Furthermore, the combinatorial tropical multiplicities of Kerber and Markwig for counts in ℙ2 are derived and explained algebro-geometrically, using Berkovich geometry and logarithmic Gromov--Witten theory. As a consequence, a new proof of Pandharipande's formula for counts of elliptic curves in ℙ2 with fixed j-invariant is obtained.PostprintPeer reviewe

Non-Archimedean Geometry and Applications

The workshop focused on recent developments in non-Archimedean analytic geometry with various applications to other fields, in particular to number theory and algebraic geometry. These applications included Mirror Symmetry, the Langlands program, p-adic Hodge theory, tropical geometry, resolution of singularities and the geometry of moduli spaces. Much emphasis was put on making the list of talks to reflect this diversity, thereby fostering the mutual inspiration which comes from such interactions