Linear Systems on Higher Dimensional Varieties

首席研究员建议研究高维变元的线性系统。特别是，他计划应用高维几何的方法来研究代数几何不同领域的一些基本问题，如二次几何、交换代数、计算复杂性、代数曲线的几何、奇异变元的喷射方案和大因子几何。他已经在以下方面获得了结果：(1)关于理想的Nullstellensz和合集的有效结果；(2)theta因子的奇性和不规则簇的双曲几何；(3)Fujita关于伴随线性系统的问题；(4)研究理想的符号权。但仍有许多根本性问题尚未解决。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。代数测地学在数学物理和密码学中也有应用。在这个项目中，首席研究员开发了新的技术来研究代数、计算复杂性和高维几何对象的性质。

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.