March 7, Wednesday, 15:30, room 211

Dmitry Korshunov

Line bundles, the degree of a projective variety, Hilbert polynomial

We will survey basic definitions and properties of line bundles on projective varieties, as well as their relations to embeddings into projective spaces. Our aim will be to find an intrinsic definition of degree of a projective variety. The computable way to do it is via Hilbert polynomial. Consider the generating function counting dimensions of homogeneous components of the coordinate ring of a variety. It turns out that for large degrees this function coincides with some well defined polynomial over rationals, containing a lot of intrinsic information about the variety. In particular the dimension and degree can be read off from the leading monomial.