Kenji Matsuki
This is an expanded version of the notes for the lectures given by the author at RIMS in the summer of 1999 to give a detailed account of the proof for the (weak) factorization theorem of birational maps by Abramovich-Karu-Matsuki-Włodarczyk.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Lecture notes of a course on birational geometry (taught at College de France, Winter 2011, with the support of Fondation Sciences Mathématiques de Paris). Topics covered: introduction into the subject, contractions and extremal rays, pairs and singularities, Kodaira dimension, minimal model program, cone and contraction, vanishing, base point freeness, flips and local finite generation, pl flips and extension theorems, existence of minimal models and Mori fibre spaces, global finite generation, etc.
This paper has been withdrawn by the author, due to a counter example to step 6.2 indicated by K. Karu.
We prove functorial weak factorization of projective birational morphisms of regular quasi-excellent schemes in characteristic 0 broadly based on the existing line of proof for varieties. From this general functorial statement we deduce factorization results for algebraic stacks, formal schemes, complex analytic germs, Berkovich analytic and rigid analytic spaces.
This is a preliminary version of the paper for the Lecture Notes Series.
The recent two proofs for the (weak) factorization theorem for birational maps, one by Włodarczyk and the other by Abramovich-Karu-Matsuki-Włodarczyk rely on the results of Morelli. The former uses the process for $π$-desingularization (the most subtle part of Morelli's combinatorial algorithm), while the latter uses the strong factorization of toric (toroidal) birational maps directly. This paper provides a coherent account of Morelli's work (and its toroidal extension) clarifying some discrepancies in the original argument.
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.
Awfully idiosyncratic lecture notes from CMI summer school in arithmetic geometry July 31-August 4, 2006. Does not include: rationality problems, techniques of the minimal model problem and much of the rest. Includes: Lecture 0: geometry and arithmetic of curves Lecture 1: Kodaira dimension and properties, rational connectendess, Lang's and Campana's conjectures. Lecture 2: Campana's program; Campana constellations framed in terms of b-divisors, to allow for a definition of Kodaira dimension directly on the base. A speculative notion of firmaments which may deserve further investigation, espec…
The strong factorization conjecture states that a proper birational map between smooth algebraic varieties over a field of characteristic zero can be factored as a sequence of smooth blowups followed by a sequence of smooth blowdowns. We prove a local version of the strong factorization conjecture for toric varieties. Combining this result with the monomialization theorem of S.D.Cutkosky, we obtain a strong factorization theorem for local rings dominated by a valuation.
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.
In this paper we develop a Morse-like theory in order to decompose birational maps and morphisms of smooth projective varieties defined over a field of characteristic zero into more elementary steps which are locally étale isomorphic to equivariant flips, blow-ups and blow-downs of toric varieties. A crucial role in the considerations is played by K^*-actions where K is the base field. This paper serves as a basis for proving the weak factorization conjecture on factorization of birational maps in characteristic zero into blow-ups and blow-downs. This is carried out in two subsequent papers, o…
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。
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.
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.
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/990407…
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.
In this note, we present the theorem of extension of birational group laws in both settings of classical varieties (Weil) and schemes (Artin). We improve slightly the original proof with a more direct construction of the group extension and the systematic use of algebraic spaces, and we discuss the separation properties of the group extension. We also explain the important application to the construction of Néron models of abelian varieties. This note grew out of lectures given by Ariane Mézard and the second author at the Summer School "Schémas en groupes" held in the CIRM (Luminy) from 29 Au…
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 $U$.
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。
This is an expanded version of the third author's lecture in String-Math 2015 at Sanya. It summarizes some of our works in quantum cohomology. After reviewing the quantum Lefschetz and quantum Leray--Hirsch, we discuss their applications to the functoriality properties under special smooth flops, flips and blow-ups. Finally, for conifold transitions of Calabi--Yau 3-folds, formulations for small resolutions (blow-ups along Weil divisors) are sketched.
These are notes of lectures given at the school `Birational Geometry of Hypersurfaces' in Gargnano in March 2018. The main goal was to discuss the Hodge structures that come naturally associated with a cubic fourfold. The emphasis is on the Hodge and lattice theoretic aspects with many technical details worked out explicitly. More geometric or derived results are only hinted at.
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.
These are the lecture notes based on earlier papers with some additional new results. New and simple proofs are given for local freeness theorem and the semipositivity theorem. A decomposition theorem for higher direct images of dualizing sheaves of Kollár is extended to the sheaves of differential forms of arbitrary degrees in the case of a well prepared birational model. We will also prove the log versions of some of the results, i.e., the case where we allow horizontal boundary components.
These notes are based on three lectures given at the 2013 CIME/CIRM summer school. The purpose of this series of lectures is to introduce the notion of a toric fibration and to give its geometrical and combinatorial characterizations. Polarized toric varieties which are birationally equivalent to projective toric bundles are associated to a class of polytopes called Cayley polytopes. Their geometry and combinatorics have a fruitful interplay leading to fundamental insight in both directions. These notes will illustrate geometrical phenomena, in algebraic geometry and neighboring fields, which …
These are the notes of the lectures delivered by the author at CIME in June 2018. The main purpose of the notes is to provide an overview of the techniques used in the construction of the triply graded link homology. The homology is space of global sections of a particular sheaf on the Hilbert scheme of points on the plane. Our construction relies on existence on the natural push-forward functor for the equivariant matrix factorizations, we explain the subtleties on the construction in these notes. We also outline a proof of the Markov moves for our homology as well as some explicit localizati…
In this Lecture Notes we present, in a sufficiently self contained way, our contributions and interests in the field of Minimal Model Theory. We study Fano-Mori spaces, both from the biregular and the birational point of view. For the former we recall and develop Kawamata's Base Point Free technique and some of Mori's deformation arguments. For the latter we lean on Sarkisov and #-Minimal Model Programs. In writing these notes we want to give our point of view on this area of research. We are not trying to give a treatment of the whole subject. These notes collect some topics we presented in t…
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。
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…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。