Weak semistable reduction in characteristic 0

1997/07/13 by Abramovich, Dan, Karu, Kalle
#14D05 #14M25 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.alg-geom/9707012

Abstract

Let X->B be a morphism of varieties in characteristic zero. Semistable reduction has been proved for dim(B)=1 (Kempf, Knudsen, Mumford, Saint-Donat), dim(X)=dim(B)-1 (de Jong) and dim(X)=dim(B)+2 (Alexeev, Kollar, Shepherd-Barron). In this paper we consider the general case. First we define what we mean by a semistable morphism in terms of toroidal embeddings. Then we reduce the varieties to toroidal embeddings and solve a slightly weaker version of semistable reduction. We also state the full semistable reduction problem in terms of combinatorics of the associated polyhedral complexes.

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