English

Smoothness, Semistability, and Toroidal Geometry

alg-geom 2015-06-30 v1 Algebraic Geometry

Abstract

We provide a new proof of the following result: Let XX be a variety of finite type over an algebraically closed field kk of characteristic 0, let ZXZ\subset X be a proper closed subset. There exists a modification f:X1\rarXf:X_1 \rar X, such that X1X_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.

Keywords

Cite

@article{arxiv.alg-geom/9603018,
  title  = {Smoothness, Semistability, and Toroidal Geometry},
  author = {Dan Abramovich and Johan de Jong},
  journal= {arXiv preprint arXiv:alg-geom/9603018},
  year   = {2015}
}

Comments

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