Factorization of Birational Maps

本课题的主要目的是研究双态的强分解猜想：在非奇异复变之间的每一个适当的双态都可以被分解为一个光滑爆炸的组合，然后是一个光滑爆炸的组合。这个猜想来自于一个更基本的问题，即用平滑膨胀法构造一个态射环面。有几种方法可以解决环化问题，例如，将其简化为具有对数极点的微分形式的奇异解问题，或者使用低维的几个结构的组合来实现高维的环化。本研究的主题是研究代数变量的分类问题。更准确地说，给定两个一般相同的变体（在除去一些小子集之后它们是相同的），可以通过一些基本操作将一个变体转换为另一个变体吗？最简单的初等运算是子变量的放大和缩小：用更大的子变量替换一个子变量。较小的尺寸。吹-吹-吹猜想指出，人们总是可以从第一种变化到第二种变化，通过几次吹起来，然后吹下来。

The main goal of the proposed project is to study the strong factorizationconjecture for birational morphisms: every proper birational morphismbetween nonsingular complex varieties can be factored as acomposition of smooth blowups followed by a composition of smoothblowdowns. The conjecture follows from the morefundamental problem of making a morphism toroidal by smoothblowups. There are several approaches to solving the toroidalizationproblem, for example, reducing it to a problem about resolution ofsingularities of a differential form with logarithmic poles, or usinga composition of several constructions in lower dimensions to achievetoroidalization in higher dimension.The main theme of the proposed research is to study the classificationof algebraic varieties. More precisely, given two varieties that arethe same generically (they are the same after removing some smallsubsets), can one transform one variety to another by some elementaryoperations? The simplest elementary operations are blowups andblowdowns of subvarieties: replacing a subvariety by anothersubvariety of larger (resp. smaller) dimension. The blowup-blowdownconjecture states that one can always get from the first variety tothe second by blowing up several times and then blowing down.