Weak semistable reduction in characteristic 0
type: Item
instance of: scholarly article
title: Weak semistable reduction in characteristic 0
zbMATH Open document ID: 0958.14006
author: Dan Abramovich
author: Kalle Karu
published in: Inventiones Mathematicae
publication date: 2000-07-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/alg-geom/9707012
review text: 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.
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 1443405
zbMATH Keywords: birational map
zbMATH Keywords: singularities
zbMATH Keywords: resolution singularities
zbMATH Keywords: connected fibers
zbMATH Keywords: weak semistable morphism
zbMATH Keywords: alteration
zbMATH Keywords: modification
reviewed by: Marcel Morales
MaRDI profile type: Publication
OpenAlex ID: W1992047631
arXiv ID: alg-geom/9707012
DOI: 10.1007/S002229900024
description: scientific article