Plan
Here is a quick summary of how I currently plan to approach this course. Please note that this does not follow the plan given in the Course Outline (which is a suggested approach to essentially the same topics)!
As most of you know, I will give notes/slides for each of my lectures rather than asking you to read a specific book. However, the book by Hartshorne has some nice exercises which are worth doing.
Base topics
Roughly speaking, this is my own approach to "Chapter 1" of Hartshorne's book on Algebraic Geometry. The idea is to try to understand the geometric meaning of algebraic notions in the context of the graded polynomial ring. This will give a preview/basis of the more formal definitions introduced later.
Projective space over a field. Linear subspaces. Homogeneous Co-ordinates. The graded co-ordinate ring.
Homogeneous systems of equations (and inequalities) and "sub-schemes" of projective space defined by them.
Graded modules and "(quasi)-Coherent sheaves" on Projective space.
"Generic" points and prime ideals. Primary decomposition.
Hilbert functions and associated homological algebra.
Examples that will be scattered through 1-4. Products of projective spaces. Affine schemes. Veronese embedding. Hypersurfaces. Curves. Quadrics. Line bundles and Vector Bundles. "Thick points".
Categorical topics
These are roughly the topics in the later chapters of Hartshorne's book.
Category of (quasi)-projective schemes. Topology and gluing. Sheaves and localisation. Schemes as ringed spaces.
Abelian categories, Complexes and Cohomology of sheaves.
Advanced topics
We may or may not do some of these topics.
Serre Duality.
Riemann-Roch for Curves.
Smoothness, regularity and Bertini's theorem.