Vol. 8, No. 2, 2014

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ISSN: 1944-7833 (e-only)
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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 OK and algebraically closed residue field k of characteristic p > 0. Let XK be a smooth proper geometrically connected curve of genus g > 0 with X(K) if g = 1. Assume that XK does not have good reduction and that it obtains good reduction over a Galois extension LK of degree p. Let YOL be the smooth model of XLL. Let H := Gal(LK).

In this article, we provide information on the regular model of XK obtained by desingularizing the wild quotient singularities of the quotient YH. The most precise information on the resolution of these quotient singularities is obtained when the special fiber Ykk 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 XK 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