Factorization of Birational Maps

双有理图因式分解

• 批准号: 0070678
• 负责人: Joseph Harris
• 金额: $ 9.1万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070678&HistoricalAwards=false
• 关键词: Factorization; Birational; Maps

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.

该项目的主要目的是研究二元态射的强因式分解猜想：非奇异复变簇之间的每一个适当的二元态射都可以分解为光滑爆破和光滑爆破的合成。这一猜想源于一个更基本的问题，即通过平滑放大使态射成为环面。环球化问题有几种解决方法，例如，将其归结为具有对数极点的微分形式的奇点分解问题，或者利用低维的几个结构的组合来实现高维的环球化。研究的主要主题是代数簇的分类。更准确地说，假设有两个变种是相同的(去掉一些小的子集后它们是相同的)，一个变种可以通过一些基本操作转换成另一个变种吗？最简单的基本操作是放大和缩小亚种：将一个亚种替换为另一个较大的亚种(分别为。较小)尺寸。吹气-吹气猜想指出，一个人总是可以通过几次吹气然后再吹下来，从第一个品种变成第二个品种。