Blowup Algebras of Ideals and Modules, and Applications

项目的第一部分旨在了解理想的爆破代数，其中包括Rees环和相关的分次环。主要目的之一是用约化数刻画这些代数的Cohen-Macaulay性。主要研究者还打算通过定义爆破代数的方程来描述它们。这与确定射影簇之间有理映射象的方程密切相关。该项目的第二部分涉及模的Rees代数及其应用。其中一个问题涉及定义理想以及各种交换矩阵的Cohen-Macaulay性和正规性。本质上，这是一个关于射影二维模的Rees代数的问题。另一个问题涉及射影簇及其局部类比的高斯映象。用模的Rees代数来表述，目标是建立微分形式模的解析展开的一个下界。理想的Rees代数是描述簇沿子簇爆破的代数对象。虽然爆破是变种研究中的一个基本操作，但对这一过程的明确理解仍然是一个悬而未决的问题。该项目的目的是对在此背景下出现的代数进行详细的调查。此外，还将研究更广泛类别的模的Rees代数，以期在代数和几何问题上有各种应用。

ULRICH, 9970504The first part of the project aims at understanding blowup algebras of ideals, which include Rees rings and associated graded rings. One of the main objectives is to characterize the Cohen-Macaulayness of these algebras in terms of reduction numbers. The principal investigator also intends to describe blowup algebras by means of the equations defining them. This is closely related to determining the equations of the image of a rational map between projective varieties. The second part of the project concerns Rees algebras of modules and their applications. One of the questions deals with the defining ideal as well as the Cohen-Macaulayness and normality of varieties of commuting matrices. In essence, this is a problem about Rees algebras of certain modules of projective dimension two. Another question concerns the Gauss image of projective varieties and local analogues thereof. Phrased in terms of Rees algebras of modules, the objective becomes to establish a lower bound for the analytic spread of the module of differential forms. Rees algebras of ideals are algebraic objects that describe the blowup of a variety along a subvariety. Although blowing up is a fundamental operation in the birational study of varieties, an explicit understanding of this process remains an open problem. The aim of the project is a detailed investigation of the algebras that arise in this context. In addition, the wider class of Rees algebras of modules will be studied, with a view towards various applications to algebraic and geometric problems.