5,603 research outputs found
Metric uniformization of morphisms of Berkovich curves
We show that the metric structure of morphisms between
quasi-smooth compact Berkovich curves over an algebraically closed field admits
a finite combinatorial description. In particular, for a large enough skeleton
of , the sets of points of of
multiplicity at least in the fiber are radial around with the
radius changing piecewise monomially along . In this case, for any
interval connecting a rigid point to the skeleton, the
restriction gives rise to a piecewise monomial function
that depends only on the type 2 point
. In particular, the metric structure of is determined by
and the family of the profile functions with
. We prove that this family is piecewise monomial in
and naturally extends to the whole . In addition, we extend
the theory of higher ramification groups to arbitrary real-valued fields and
show that coincides with the Herbrand's function of
. 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
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