  In the famous paper of Deligne and Mumford, they proved that a proper
hyperbolic curve over a discrete valuation field has stable reduction if and
only if the Jacobian variety of the curve has stable reduction in the case
where the residue field of its valuation ring is algebraically closed. In the
proof, the theory of minimal regular models played an important role. In this
paper, we establish a theory of minimal log regular models of curves. As a key
tool for this theory, we give a criterion for $2$-dimensional local schemes to
be log regular in terms of their minimal desingularization. Moreover, as an
application of this theory, we prove the above equivalence without the
assumption on the residue field.
