Prime Characteristic Techniques in Commutative Algebra and Algebraic Geometry

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一个光滑吹气和一个光滑吹气的合成。这一猜想源于一个更基本的问题，即通过平滑膨胀使态射成为环状。有几种方法可以解决环面问题，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面问题。更准确地说，假设两个变种在一般情况下是相同的(除去一些小的子集后它们是相同的)，一个变种可以通过一些初等运算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：用更大的亚种的另一个亚种替换一个亚种。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。

The main goal of the proposed project is to study the strong factorization conjecture for birational morphisms: every proper birational morphism between nonsingular complex varieties can be factored as a composition of smooth blowups followed by a composition of smooth blowdowns. The conjecture follows from the more fundamental problem of making a morphism toroidal by smooth blowups. There are several approaches to solving the toroidalization problem, for example, reducing it to a problem about resolution of singularities of a differential form with logarithmic poles, or using a composition of several constructions in lower dimensions to achieve toroidalization in higher dimension.The main theme of the proposed research is to study the classification of algebraic varieties. More precisely, given two varieties that are the same generically (they are the same after removing some small subsets), can one transform one variety to another by some elementary operations? The simplest elementary operations are blowups and blowdowns of subvarieties: replacing a subvariety by another subvariety of larger (resp. smaller) dimension. The blowup-blowdown conjecture states that one can always get from the first variety to the second by blowing up several times and then blowing down.