Blowup Algebras of Ideals and Modules, and Applications

点击翻译按钮获取中文摘要

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.