Linear Systems on Higher Dimensional Varieties

主要研究者建议研究高维变量的线性系统。特别是，他计划应用高维几何的方法来研究一些基本问题，从不同领域的代数几何，如双有理几何，交换代数，计算复杂性，几何的代数曲线，喷气计划的奇异品种和几何的大因子。他已经取得了成果，在以下领域，（1）有效的结果nullstellenandsyzygies的理想，（2）奇异的θ因子和双理性几何的不规则品种，（3）藤田的问题，伴随线性系统和（4）研究象征权力的理想。但仍有许多根本性问题没有解决。他计划开发新的方法和技术来研究它们。代数几何是数学中最古老的领域之一，它研究由代数方程定义的几何对象的几何，代数和算术性质。代数地球测量学也应用于数学物理学和密码学。在这个项目中，首席研究员开发新的技术来研究代数，计算复杂性和高维几何对象的属性。

The principal investigator proposes to study linear systems of higher dimensional varieties. In particular, he plans to apply the methods of higher dimensional geometry to study some basic questions from different areas of algebraic geometry such as birational geometry, commutative algebra, computational complexity, geometry of algebraic curves, jet schemes of singular varieties and geometry of big divisors. He has already obtained results in the following areas, (1) effective results onNullstellensatz and syzygies of ideals, (2) singularities of theta divisors and birational geometry of irregular varieties, (3) Fujita's problems on adjoint linear systems and (4) studying symbolic power of an ideal. But there are still many fundamental problems remain unsolved. He plans to develop new methods and techniques to study them.Algebraic geometry is one of the oldest areas in mathematics and it studies geometric, algebraic and arithmetic properties of geometric objects defined by algebraic equations. Algebraic geoemtry also has applications to mathematical physics and cryptography. In this project, the principal investigator develops new techniques to study algebra, computational complexity and properties of higher dimensional geometric objects.