Local & Global Problems

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9400753 Cutkosky The principle investigator will investigate four problems in algebraic geometry. He will work on the Riemann-Roch problem which is to compute the cohomology groups of high multiples of a line bundle on a projective variety. He will also work on the local analogue of this problem which is the computation of Hilbert functions of valuation ideals determined by vanishing to high multiplicity along fixed valuations. In addition, the principle investigator will study ramification and the equivalence of homomorphisms of modules. This research is concerned with a number of questions in commutative algebra and algebraic geometry. Algebraic geometry studies solutions of families of polynomial equations. One can either study the geometry of the solution set or approach problems algebraically by investigating certain functions on the solution set that form what is called a commutative ring. This dual perspective creates a close connection between commutative algebra and algebraic geometry.