Termination and Vector Bundles on Projective Space

首席调查员将对代数变种的分类进行研究。代数几何研究多项式方程。这个主题始于古希腊人的工作，他们考虑圆锥体的部分；这些部分是圆、椭圆、抛物线和双曲线。代数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表面的问题。总是可以只使用两个方程式吗？如何在不参考方程的情况下本质地描述几何。PI将致力于Hartthene关于射影空间的两个子簇是完全交的光滑余维的猜想。通过Serre的工作，这等价于具有大于或等于七个分裂的维度的射影空间上的每个二阶向量丛是线丛的直和。向量束有两种类型，不稳定和稳定，PI将分别在每个部分上工作。PI还将对Shokurov的一个猜想进行研究，该猜想与原木差异的升链条件有关，是证明翻转终止的一大步。这一裁决反映了NSF的法定使命，并通过使用基金会的智力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.