Metric uniformization of morphisms of Berkovich curves

We show that the metric structure of morphisms $f\colon Y\to X$ between quasi-smooth compact Berkovich curves over an algebraically closed field admits a finite combinatorial description. In particular, for a large enough skeleton $\Gamma=(\Gamma_Y,\Gamma_X)$ of $f$, the sets $N_{f,\ge n}$ of points of $Y$ of multiplicity at least $n$ in the fiber are radial around $\Gamma_Y$ with the radius changing piecewise monomially along $\Gamma_Y$. In this case, for any interval $l=[z,y]\subset Y$ connecting a rigid point $z$ to the skeleton, the restriction $f|_l$ gives rise to a $profile$ piecewise monomial function $\varphi_y\colon [0,1]\to[0,1]$ that depends only on the type 2 point $y\in\Gamma_Y$. In particular, the metric structure of $f$ is determined by $\Gamma$ and the family of the profile functions $\{\varphi_y\}$ with $y\in\Gamma_Y^{(2)}$. We prove that this family is piecewise monomial in $y$ and naturally extends to the whole $Y^{\mathrm{hyp}}$. In addition, we extend the theory of higher ramification groups to arbitrary real-valued fields and show that $\varphi_y$ coincides with the Herbrand's function of $\mathcal{H}(y)/\mathcal{H}(f(y))$. This gives a curious geometric interpretation of the Herbrand's function, which applies also to non-normal and even inseparable extensions.Comment: second version, 28 page

Stable modification of relative curves

We generalize theorems of Deligne-Mumford and de Jong on semi-stable modifications of families of proper curves. The main result states that after a generically \'etale alteration of the base any (not necessarily proper) family of multipointed curves with semi-stable generic fiber admits a minimal semi-stable modification. The latter can also be characterized by the property that its geometric fibers have no certain exceptional components. The main step of our proof is uniformization of one-dimensional extensions of valued fields. Riemann-Zariski spaces are then used to obtain the result over any integral base.Comment: 60 pages, third version, the paper was revised due to referee's report, section 2 was divided into sections 2 and 6, to appear in JA

Wild coverings of Berkovich curves

This paper is an extended version of the author's talk given at the conference "Non-Archimedean analytic geometry: theory and practice" held in August 2015 at Papeete. It gives a brief overview of recent results on the structure of wild coverings of Berkovich curves and its relation to the different and higher ramification theory.Comment: 8 page

Inseparable local uniformization

It is known since the works of Zariski in early 40ies that desingularization of varieties along valuations (called local uniformization of valuations) can be considered as the local part of the desingularization problem. It is still an open problem if local uniformization exists in positive characteristic and dimension larger than three. In this paper, we prove that Zariski local uniformization of algebraic varieties is always possible after a purely inseparable extension of the field of rational functions, i.e. any valuation can be uniformized by a purely inseparable alteration.Comment: 66 pages, final version, the paper was seriously revise