Termination and Vector Bundles on Projective Space

首席研究员将对代数变量的分类进行研究。代数几何涉及多项式方程的研究。这个主题始于古希腊人的工作，他们考虑圆锥体的部分；这些是圆、椭圆、抛物线和双曲线。代数上，二次曲线对应于二次方程。PI将考虑寻找最方便的方法来描述四维空间中的曲面的问题。总是可以只用两个方程吗？不借助方程，怎么能从本质上描述几何呢？PI将研究Hartshorne关于光滑余维投影空间的两个子簇是完全相交的猜想。根据Serre的工作，这等价于这样的陈述：在一个维数大于或等于7的射影空间中，每一个二阶向量束都是直线束的直接和。矢量束有两种类型，不稳定和稳定，PI将分别在每一块上工作。PI还将研究Shokurov的一个猜想，该猜想与对数差异的上升链条件有关，这是证明翻转终止的一大步。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The Principal Investigator will conduct research on the classification of algebraic varieties. Algebraic Geometry concerns the study of polynomial equations. The subject started with the work of the ancient Greeks, who considered sections of a cone; these are circles ellipses, parabolas and hyperbolas. Algebraically conic sections correspond to quadratic equations. The PI will consider the problems of finding the most convenient way to describe surfaces in four dimensional space. Is it always possible to use only two equations? How can one describe the geometry intrinsically, without reference to equations.The PI will work on Hartshorne's conjecture on smooth codimension two subvariety of projective spaces being a complete intersection. By work of Serre this is equivalent to the statement that every rank two vector bundle on a projective space having dimension greater or equal than seven splits as a direct sum of line bundles. Vector bundles come in two types, unstable and stable and the PI will work on each piece separately. The PI will also work on a conjecture of Shokurov to do with the ascending chain condition for the log discrepancy and which is a big step toward proving termination of flips.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.