Commutative Algebraic Aspects of Birational Algebraic Geometry

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This project focus on problems at the interface of commutative algebra andalgebraic geometry. Specifically, several commutatative algebraicapproaches to problems in higher dimensional birational algebraic geometryare proposed. These include building on the PI's prior work on ideal coresto prove a conjecture of Kawamata's on the existence of a global sectionfor any ample line bundle on a smooth projective variety which is adjointto an ample bundle. It also includes the development of the theory ofmultiplier ideals in the singular setting, and a study of the commutativealgebraic properties of jet schemes, especially for combinatorially richcases 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 studythe geometric objects (called algebraic varieties) that can be describedby polynomial equations. These fields provide the underpinnings of severalimportant areas of applications of mathematics to problems in governmentand industry. Examples include error-correcting codes based on algebraicvarieties, and computer-aided design (CAD) which "draws" geometric objectson the computer screen by plotting pieces of algebraic varieties. Whilethis project does not address specific current problems in applications,it does contribute to the healthy thriving infrastructure of the subjectnecessary to support applications in the future. Furthermore, it isexpected that several of trainees in this project will continue later withmore applied aspects of the subject by moving to positions in industry andgovernment (such as the National Security Agency).