Dedicated to John Tate Abstract: Let K be a complete discrete valuation field with ring of integers OK and residue field k of characteristic p ≥ 0, assumed to be algebraically closed. Let X/K denote a smooth proper geometrically connected curve of genus g ≥ 1, and let X/OK denote its minimal regular model. When g ≥ 2, or g = 1 and X(K) 6= ∅, there exists a finite Galois extension L/K minimal with the property that XL/L has semi-stable reduction. Let X ′/OL denote the minimal regular model of XL/L. We discuss in this article properties of the special fiber of X ′ and of the extension L/K that can be inferred from the knowledge of the combinatorial properties of the special fiber of X .