A Note on Stable Reduction of Smooth Curves Whose Jacobians Admit Stable Reduction

Abstract

P. Deligne and D. Mumford proved that, for a smooth curve over the field of fractions of a discrete valuation ring whose residue field is perfect, if the associated Jacobian has stable reduction over the discrete valuation ring, then the smooth curve has stable reduction over the discrete valuation ring. Recently, I. Nagamachi proved a similar result over a connected normal Noetherian scheme of dimension one. In the present paper, we prove a similar result over a Prüfer domain, i.e., a domain whose localization at each of the prime ideals is a valuation ring. Moreover, we also give a counter-example in a situation over a higher dimensional base case. More precisely, we construct an example of a smooth curve over the field of fractions of a complete strictly Henselian normal Noetherian local domain of equal characteristic zero such that the associated Jacobian has good reduction over the local domain, but the smooth curve does not have stable reduction over the local domain.