Torification and Factorization of Birational Maps

Erratum to the paper "A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension"

This is an erratum to math.AG/9803126, Tohoku 51 (1999) 489-537. This erratum describes: 1. the failure of the algorithm in [AMR] and [Morelli1] for the strong factorization pointed out by Kalle Karu, 2. the statement of a refined weak factorization theorem for toroidal birational morphisms in [AMR], in the form utilized in [AKMR] for the proof of the weak factorization theorem for general birationla maps, avoiding the use of the above mentioned algorithm for the strong factorization, and 3. a list of corrections for a few other mistakes in [AMR], mostly pointed out by Laurent Bonavero.

Factorizations of birational extensions of local rings

We give a proof of local strong factorization of a birational extension of regular local rings (of equicharacteristic zero) along a valuation of rank 1 and maximal rational rank. This gives an alternate proof to the geometric proof of this Theorem by Karu. Our proof uses only linear algebra, and is in the spirit of Christensen's proof of this result in dimension 3.

二维情形下的分解中心与格罗滕迪克簇环

Factorization centers in dimension two and the Grothendieck ring of varieties

本文研究双有理映射的分解中心，证明完美域上光滑射影曲面的双有理自同构的弱分解中胀开与胀缩中心同构，由此推出无法用曲面双有理自同构构造0维簇的非平凡L-等价，且有理曲面的有理性中心是良定义的。

Simple Constructive Weak Factorization

We give a simplified algorithm of the functorial weak factorization of birational morphisms of nonsingular varieties over a field of characteristic zero into a composite of blow-ups and blow-downs with smooth centers.

On the birational geometry of toric Fano 4-folds

We announce a factorization result for equivariant birational morphisms between toric 4-folds whose source is Fano: such a morphism is always a composite of blow-ups along smooth invariant centers. Moreover, we show with a counterexample that, differently from the 3-dimensional case, even if both source and target are Fano, the intermediate varieties can not be chosen Fano.

Factorization Theorem for Projective Varieties with Finite Quotient Singularities

In this paper, we prove that any two birational projective varieties with finite quotient singularities can be realized as two geometric GIT quotients of a non-singular projective variety by a reductive algebraic group. Then, by applying the theory of Variation of Geometric Invariant Theory Quotients ([3]), we show that they are related by a sequence of GIT wall-crossing flips.

A GIT proof of Włodarczyk's weighted factorization theorem

We give a short proof of W{\lodarczyk's theorem that any birational map between smooth projective varieties in characteristic zero is a composition of weighted blowups and blowdowns.

Factorization for stacks and boundary complexes

We prove a weak factorization result on birational maps of Deligne-Mumford stacks, and deduce the following: Let $U \subset X$ be an open embedding of smooth Deligne-Mumford stacks such that $D = X-U$ is a normal crossings divisor, then the the simple homotopy type of the boundary complex $Δ(X,D)$ depends only on $U$.

构造Mori Dream Spaces之间双有理映射的几何实现

Constructing geometric realizations of birational maps between Mori Dream Spaces

本文为Mori Dream Spaces间的双有理映射构造几何实现，证明其仍为MDS且诱导墙穿越分解，并提供SageMath工具及环面情形下的矩多胞体构造。

基于关于吹胀的多项式指标的双有理映射的动力次数 II. 三维示例

Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups II. 3D examples

本文提出一种利用多项式指标与拉回因式分解计算双有理映射迭代次数的新方法，通过三维示例验证其灵活性，无需构造代数稳定提升，优于传统皮卡群方法。

Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups I. General theory and 2D examples

In this paper we address the problem of computing $\text{deg}(f^n)$, the degrees of iterates of a birational map $f:\mathbb{P}^N\rightarrow\mathbb{P}^N$. For this goal, we develop a method based on two main ingredients: the factorization of a polynomial under pull-back of $f$, based on local indices of a polynomial associated to blow-ups used to resolve the contraction of hypersurfaces by $f$, and the propagation of these indices along orbits of $f$. For maps admitting algebraically stable modifications $f_X:X\rightarrow X$, where $X$ is a variety obtained from $\mathbb P^N$ by a finite number…

Integral Grothendieck-Riemann-Roch theorem

We show that, in characteristic zero, the obvious integral version of the Grothendieck-Riemann-Roch formula obtained by clearing the denominators of the Todd and Chern characters is true (without having to divide the Chow groups by their torsion subgroups). The proof introduces an alternative to Grothendieck's strategy: we use resolution of singularities and the weak factorization theorem for birational maps.