250116 VO Graded Rings and Projective Geometry (2023S)

The purpose of this course is to introduce methods to study projective varieties in (weighted) projective space via looking at the algebraic structure of the associated graded coordinate ring. Numerical invariants of the ring (its Hilbert function, so-called Hilbert numerator, etc) are related to the underlying space via Riemann-Roch type theorems. Algebraic structures of the ring (whether it is Cohen-Macaulay, Gorenstein, etc) are reflected in geometric properties of the embedding. The course will be very much examples-based; we will recall, and build up, the theory as we go along.