## Algebra & Number Theory

### Wild models of curves

Dino Lorenzini

#### Abstract

Let $K$ be a complete discrete valuation field with ring of integers $OK$ 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 $X(K)≠∅$ 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 $Y∕OL$ be the smooth model of $XL∕L$. Let $H:= Gal(L∕K)$.

In this article, we provide information on the regular model of $X∕K$ obtained by desingularizing the wild quotient singularities of the quotient $Y∕H$. The most precise information on the resolution of these quotient singularities is obtained when the special fiber $Yk∕k$ 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 $X$.

#### Article information

Source
Algebra Number Theory, Volume 8, Number 2 (2014), 331-367.

Dates
Revised: 6 June 2013
Accepted: 16 July 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513730153

Digital Object Identifier
doi:10.2140/ant.2014.8.331

Mathematical Reviews number (MathSciNet)
MR3212859

Zentralblatt MATH identifier
1332.14029

#### Citation

Lorenzini, Dino. Wild models of curves. Algebra Number Theory 8 (2014), no. 2, 331--367. doi:10.2140/ant.2014.8.331. https://projecteuclid.org/euclid.ant/1513730153

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