Factorization of Birational Maps

该项目的主要目标是研究双有理态射的强分解猜想：非奇异复簇之间的每一个真双有理态射都可以分解为光滑blowup的合成，然后是光滑blowdown的合成。这个猜想来自于一个更基本的问题，即通过光滑的爆破使一个态射成为一个环形。解决超环面化问题有几种方法，例如，将其归结为对数极点的微分形式的奇异性的分解问题，或者用低维的几个结构的组合来解决高维的超环面化问题。更确切地说，给定两个类属相同的变种（去掉一些小子集后它们是相同的），人们能通过一些基本运算将一个变种转换成另一个变种吗？最简单的初等运算是子簇的blowup和blowdown：用更大的子簇替换一个子簇。较小）尺寸。blowup-blowdown猜想指出，人们总是可以通过多次blowup然后blowdown来从第一个变量得到第二个变量。

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.