Semi-stable reduction and models of curves

Abstract

Let S be a Dedekind scheme of dimension 1 and let X be a smooth, projective, geometrically connected curve of genus g>=2 over the function field K(S) of S. The semi-stable reduction theorem tells us that we can find a field extension K' of K(S) such that the base change X' of X over K' has a semi-stable model over S', the normalization of S in K'. The first part of the thesis treats some of the theory necessary to understand the proof of Artin-Winters of the semi-stable reduction theorem.

In the second part of the thesis a bound is found for the degree of the necessary field extension needed to achieve semi-stable reduction. After that it is shown how one can use the semi-stable reduction theorem to lift a separable tamely ramified cover of curves to a finite morphism of semi-stable models of curves after a finite field extension. A bound on the degree of the necessary field extension to achieve this is also found.