Termination and Vector Bundles on Projective Space

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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.