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Title: A Note on Stable Reduction of Smooth Curves Whose Jacobians Admit Stable Reduction
Authors: HOSHI, Yuichiro
Keywords: 14H10
smooth curve
stable curve
stable reduction
Issue Date: Aug-2022
Publisher: Research Institute for Mathematical Sciences, Kyoto University
Start page: 1
End page: 10
Thesis number: RIMS-1964
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.
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Appears in Collections:Research Institute for Mathematical Sciences, preprints

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