Néron models of algebraic curves

Abstract

Let $S$ be a Dedekind scheme with field of functions $K$. We show that if $X_K$ is a smooth connected proper curve of positive genus over $K$, then it admits a Néron model over $S$, i.e., a smooth separated model of finite type satisfying the usual Néron mapping property. It is given by the smooth locus of the minimal proper regular model of $X_K$ over $S$, as in the case of elliptic curves. When $S$ is excellent, a similar result holds for connected smooth affine curves different from the affine line, with locally finite type Néron models.