#### Vol. 8, No. 2, 2014

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Wild models of curves

### Dino Lorenzini

Vol. 8 (2014), No. 2, 331–367
##### Abstract

Let $K$ be a complete discrete valuation field with ring of integers ${\mathsc{O}}_{K}$ and algebraically closed residue field $k$ of characteristic $p>0$. Let $X∕K$ be a smooth proper geometrically connected curve of genus $g>0$ with $X\left(K\right)\ne \varnothing$ if $g=1$. Assume that $X∕K$ does not have good reduction and that it obtains good reduction over a Galois extension $L∕K$ of degree $p$. Let $\mathsc{Y}∕{\mathsc{O}}_{L}$ be the smooth model of ${X}_{L}∕L$. Let $H:=Gal\left(L∕K\right)$.

In this article, we provide information on the regular model of $X∕K$ obtained by desingularizing the wild quotient singularities of the quotient $\mathsc{Y}∕H$. The most precise information on the resolution of these quotient singularities is obtained when the special fiber ${\mathsc{Y}}_{k}∕k$ is ordinary. As a corollary, we are able to produce for each odd prime $p$ an infinite class of wild quotient singularities having pairwise distinct resolution graphs. The information on the regular model of $X∕K$ also allows us to gather insight into the $p$-part of the component group of the Néron model of the Jacobian of $X$.

##### Keywords
model of a curve, ordinary curve, cyclic quotient singularity, wild ramification, arithmetical tree, resolution graph, component group, Néron model
##### Mathematical Subject Classification 2010
Primary: 14G20
Secondary: 14G17, 14K15, 14J17
##### Milestones
Received: 3 January 2013
Revised: 6 June 2013
Accepted: 16 July 2013
Published: 18 May 2014
##### Authors
 Dino Lorenzini Department of Mathematics University of Georgia Athens, GA 30602 United States