Weak semistable reduction in characteristic 0
Let \(f:X\to B\) be a flat morphism of nonsingular projective varieties with connected fibers. \(f\) is called weak semistable if: (1) the varieties \(X\) and \(B\) admit toroidal structures \(U_X\subset X\) and \(U_B \subset B\) with \(U_X=f^{-1}U_B\); (2) with this structure, the morphism \(f\) is toroidal; (3) \(f\) is equidimensional; (4) all the fibers of \(f\) are reduced; and (5) \(B\) is non singular. The main theorem in this paper is the following: Let \(f:X\to B\) be a surjective morphism of projective varieties with geometrically integral generic fiber. There exist a projective alteration (that is a generically finite proper surjective morphism) \(B_1\to B\) and a projective strict modification (that is a proper birational morphism) \(Y\to X\times_BB_1\) such that \(Y\to B_1\) is weak semistable.
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