Log smooth extension of a family of curves and semi-stable reduction

Summery: We show that a family of smooth stable curves defined on the interior of a log regular scheme is extended to a log smooth scheme over the whole log regular scheme, if it is so at each generic point of the boundary, under a very mild assumption. We also include a proof of the fact that a log smooth scheme over a discrete valuation ring has potentially a semi-stable model. As a consequence, we show that a hyperbolic polycurve in the sense of [10] over a discrete valuation field has potentially a proper semi-stable model if the characteristic of the residue field is sufficiently large.

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