Smoothness, Semistability, and Toroidal Geometry

Details Smoothness, Semistability, and Toroidal Geometry Smoothness, Semistability, and Toroidal Geometry

1996-03-24

arxiv.org/abs/alg-geom/9603018v1

Mathematics

Algebraic Geometry

12 pages (in large font)., LATEX 2e (in latex 2.09 compatibility mode)

Scientific paper

We provide a new proof of the following result: Let $X$ be a variety of finite type over an algebraically closed field $k$ of characteristic 0, let $Z\subset X$ be a proper closed subset. There exists a modification $f:X_1 \rar X$, such that $X_1$ is a quasi-projective nonsingular variety and $Z_1 = f^{-1}(Z)_\red$ is a strict divisor of normal crossings. Needless to say, this theorem is a weak version of Hironaka's well known theorem on resolution of singularities. Our proof has the feature that it builds on two standard techniques of algebraic geometry: semistable reduction for curves, and toric geometry. Another proof of the same result was discovered independently by F. Bogomolov and T. Pantev. The two proofs are similar in spirit but quite different in detail.

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