Smoothness, Semistability, and Toroidal Geometry

Abstract

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.

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

Cited by in corpus (21)

• Resolution of Singularities -- Seattle Lecture
• Equivariant resolution of singularities in characteristic 0
• Toward resolution of singularities over a field of positive characteristic Part I. Foundation of the program: the language of the idealistic filtration
• Log abundance of the moduli b-divisors of lc-trivial fibrations
• Functorial destackification and weak factorization of orbifolds
• Classes of some hypersurfaces in the Grothendieck ring of varieties
• Resolution of Singularities for 3-folds in positive characteristic
• Toroidal varieties and the weak Factorization Theorem
• Torification and Factorization of Birational Maps
• A Proof of Desingularization over fields of characteristic zero
• Stable modification of relative curves
• Equisingular resolution with SNC fibers and combinatorial type of varieties
• Effective Hironaka resolution and its Complexity (with appendix on applications in positive characteristic)
• A strong desingularization theorem
• Weak Hironaka theorem
• On the geometry of p-typical covers in characteristic p
• Simple Hironaka resolution in characteristic zero
• A Newtonian and Weierstrassian Approach to Local Resolution of Singularities in Characteristic Zero
• Resolution of singularities of complex algebraic varieties and their families
• Torification of diagonalizable group actions on toroidal schemes
• Geometry of Hermitian Algebraic Functions. Quotients of Squared Norms