Commutative Algebraic Aspects of Birational Algebraic Geometry

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

This project focus on problems at the interface of commutative algebra and algebraic geometry. Specifically, several commutatative algebraic approaches to problems in higher dimensional birational algebraic geometry are proposed. These include building on the PI's prior work on ideal cores to prove a conjecture of Kawamata's on the existence of a global section for any ample line bundle on a smooth projective variety which is adjoint to an ample bundle. It also includes the development of the theory of multiplier ideals in the singular setting, and a study of the commutative algebraic properties of jet schemes, especially for combinatorially rich cases such as monomial schemes. The work proposed here is part of the broader landscape of research in "pure" algebraic geometry and commutative algebra, the fields that study the geometric objects (called algebraic varieties) that can be described by polynomial equations. These fields provide the underpinnings of several important areas of applications of mathematics to problems in government and industry. Examples include error-correcting codes based on algebraic varieties, and computer-aided design (CAD) which "draws" geometric objects on the computer screen by plotting pieces of algebraic varieties. While this project does not address specific current problems in applications, it does contribute to the healthy thriving infrastructure of the subject necessary to support applications in the future. Furthermore, it is expected that several of trainees in this project will continue later with more applied aspects of the subject by moving to positions in industry and government (such as the National Security Agency).