Stable reduction of finite covers of curves

Let K be the function field of a connected regular scheme S of dimension 1, and let f : X -> Y be a finite cover of projective smooth and geometrically connected curves over K with g(X) greater or equal to 2. Suppose that f can be extended to a finite cover {\mathcal X} -> {\mathcal Y} of semi-stable models over S (it is known that this is always possible up to finite separable extension of K). We prove that there then exists a (unique) minimal such cover. This gives a canonical way to extend X -> Y to a finite cover of semi-stable models over S.

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