Semi-stable Reduction for Curves

Given a normal proper and geometrically connected curve C over the function field K(S) of S, we seek “good” models over S. For instance, if we choose a projective embedding of C into some PK(S) then its Zariski closure in PS is an arithmetic surface (where the S-flatness uses that S is Dedekind). Hence, we can always find an arithmetic surface with generic fiber C. Since S is normal and C is reduced and geometrically connected over K(S), so H0(C,O) is a finite purely inseparable extension field of K(S), it follows from considerations with Stein factorization (whose fibers are always geometrically connected) that any such arithmetic surface has geometrically connected fibers over S. For such an arithmetic surface, we need to resolve its singularities.