Valuation Rings and Local Factorization

点击翻译按钮获取中文摘要

Cutkosky 9610132 The principal investigator will continue a study of local factorization of mappings. The investigator has recently proven that factorization of birational maps along a valuation is possible in dimension three and characteristic zero by constructing a triangle of monoidal transforms with nonsingular centers. In general, it is necessary that both rings be blownup. Some new directions which the principal investigator is pursuing are: (1) Find a more "canonical" algorithm, which will patch to a global factorization by monoidal transforms; (2) Extend the result to positive characteristic; (3) Find a local result valid in all dimensions. He is also continuing research on equivalence and finite determinancy of mappings, and on Cohen-Macaulay coordinate rings. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.