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In the famous paper of Deligne and Mumford, they proved that a properhyperbolic curve over a discrete valuation field has stable reduction if andonly if the Jacobian variety of the curve has stable reduction in the casewhere the residue field of its valuation ring is algebraically closed. In theproof, the theory of minimal regular models played an important role. In thispaper, we establish a theory of minimal log regular models of curves. As a keytool for this theory, we give a criterion for $2$-dimensional local schemes tobe log regular in terms of their minimal desingularization. Moreover, as anapplication of this theory, we prove the above equivalence without theassumption on the residue field.

Minimal log regular models of hyperbolic curves over discrete valuation fields