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- W2281745679 abstract "We show that the metric structure of morphisms $fcolon Yto X$ 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 $varphi_ycolon [0,1]to[0,1]$ that depends only on the type 2 point $yinGamma_Y$. In particular, the metric structure of $f$ is determined by $Gamma$ and the family of the profile functions ${varphi_y}$ with $yinGamma_Y^{(2)}$. 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." @default.
- W2281745679 created "2016-06-24" @default.
- W2281745679 creator A5048293958 @default.
- W2281745679 date "2014-10-25" @default.
- W2281745679 modified "2023-09-27" @default.
- W2281745679 title "Metric uniformization of morphisms of Berkovich curves" @default.
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