Building on the work of the fourth author in math.AG/9904074, we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blowings up and blowings down with smooth centers. Such a factorization exists which is functorial with respect to absolute isomorphisms, and compatible with a normal crossings divisor. The same holds for algebraic and analytic spaces. Another proof of the main theorem by the fourth author appeared in math.AG/9904076.

Torification and Factorization of Birational Maps

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Torification and factorization of birational maps

This paper completes the classic and modern results on classification of conjugacy classes of finite subgroups of the group of birational automorphisms of the complex projective plane.

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Finite Subgroups of the Plane Cremona Group

This paper is a handyman’s manual for learning how to resolve the singularities of algebraic varieties defined over a field of characteristic zero by sequences of blowups. Three objectives: Pleasant writing, easy reading, good understanding. One topic: How to prove resolution of singularities in characteristic zero. Statement to be proven (No-Tech): The solutions of a system of polynomial equations can be parametrized by the points of a manifold. Statement to be proven (Low-Tech): The zero-setX of finitely many real or complex polynomials in n variables admits a resolution of its singularities (we understand by singularities the points where X fails to be smooth). The resolution is a surjective differentiable map ε from a manifold X̃ to X which is almost everywhere a diffeomorphism, and which has in addition some nice properties (e.g., it is a composition of especially simple maps which can be explicitly constructed). Said differently, ε parametrizes the zero-set X (see Figure 1). Figure 1. Singular surface Ding-dong: The zero-set of the equation x + y = (1 − z)z in R can be parametrized by R via (s, t)→ (s(1− s) · cos t, s(1− s) · sin t, 1− s). The picture shows the intersection of the Ding-dong with a ball of radius 3. Received by the editors June 25, 2002, and, in revised form, December 3, 2002. 2000 Mathematics Subject Classification. Primary 14B05, 14E15, 32S05, 32S10, 32S45. Supported in part by FWF-Project P-15551 of the Austrian Ministry of Science. c ©2003 American Mathematical Society

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The Hironaka theorem on resolution of singularities (Or: A proof we always wanted to understand)

We present a concise proof for the existence and construction of a strong resolution of excellent schemes of finite type over a field of characteristic zero. Our proof is based on earlier work of Villamayor, Encinas-Villamayor and Bierstone-Milman. It proposes some substantial simplifications which may be helpful for a better understanding of how to prove Hironaka's famous theorem on embedded resolution of singularities.

https://www.ems-ph.org/journals/show_pdf.php?issn=0010-2571&vol=77&iss=4&rank=7

Strong resolution of singularities in characteristic zero

A new proof of equivariant resolution of singularities under a finite group action in characteristic 0 is provided. We assume we know how to resolve singularities without group action. We first prove equivariant resolution of toroidal singularities. Then we reduce the general case to the toroidal case.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1997/0004/0003/MRL-1997-0004-0003-a011.pdf

Jianhua Wang

Papers

Equivariant resolution of singularities in characteristic 0

Equivariant resolution of singularities in characteristic 0