MATH 8320: Algebraic geometry, 2nd semester

This is going to be a geometry-oriented course. Rather than studying objects in complete generality (arbitrary schemes, scheaves, etc.) and going very deeply into technical details (construction of cohomologies as derived functors, general homological algebra, etc.) I would like to help you learn some concrete algebraic geometry, such as:

What does one know about algebraic curves?
Surfaces: ruled surfaces, abelian surfaces, K3 surfaces, etc. Basic birational geometry of surfaces. Classification.
Abelian varieties; Jacobians, Picard variety, Albanese variety.
Grassmannians and flag varieties.
Threefolds and higher-dimensional varieties.

So I want to work on concrete examples and the geometric intuition, and once you have that we can do the most general case, towards the end of the semester or, better still, there will be a dedicated course on schemes, sheaves and homological algebra in the Fall. We will begin, however, with some basics which you are still missing after the first semester:

Divisors and line bundles; linear systems and maps to projective spaces.
Differentials forms and canonical class.
Riemann-Roch theorem for curves.
Blowups and birational maps.
Basics on sheaves and cohomology.

We will do this following Chapters 3 and 4 of Shafarevich, reinforced by Hartshorne.