Wild models of curves

Abstract

Let $K$ be a complete discrete valuation field with ring of integers ${\mathcal{O}}_K$ 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 $\mathcal{Y}∕{\mathcal{O}}_L$ 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 $\mathcal{Y}∕H$. The most precise information on the resolution of these quotient singularities is obtained when the special fiber ${\mathcal{Y}}_k∕k$ 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 $X$.