Research Statement: Noncommutative Projective Geometry and Vanishing Theorems

1.1. Dissertation. In the past ten years a study of “noncommutative projective geometry” has flourished. By using and generalizing techniques of commutative projective geometry, one can study certain noncommutative rings and obtain results for which no purely algebraic proof is known. The most basic building block of the theory is the twisted homogeneous coordinate ring. Let X be a projective scheme over an algebraically closed field k with σ a scheme automorphism and let L be an invertible sheaf on X. In [3] a twisted version of the homogeneous coordinate ring B = B(X,σ,L) of X was invented with the grading B = ⊕Bm for Bm = H(X,L ⊗ L ⊗ · · · ⊗ L m−1 )