Blowup Algebras of Ideals and Modules, and Applications

该项目的第一部分旨在理解理想的爆破代数，其中包括Rees环和相关的分级环。主要目标之一是用约简数来描述这些代数的Cohen-Macaulayness。主要研究者还打算用定义爆破代数的方程来描述它们。这与确定投影变量之间的有理映射的象的方程密切相关。项目的第二部分涉及模块的Rees代数及其应用。其中一个问题涉及可交换矩阵的定义理想以及各种可交换矩阵的Cohen-Macaulayness和正态性。本质上，这是一个关于某些射影维数为2的模的Rees代数的问题。另一个问题涉及投影变体的高斯象及其局部类似物。用模的Rees代数表述，目标变成了建立微分形式模的解析扩展的下界。理想的Rees代数是描述一个变量沿着一个子变量膨胀的代数对象。虽然膨胀是品种遗传研究中的一项基本操作，但对这一过程的明确理解仍然是一个悬而未决的问题。该项目的目的是在这种情况下出现的代数的详细调查。此外，将研究更广泛的Rees代数模块，并着眼于代数和几何问题的各种应用。

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.