Hartshorne, Algebraic geometry.
A year long sequence in graduate algebra, e.g., GRV-I and II. If you are
taking the GRV sequence in parallel, you should be able to get started
with this course, but may need to do some extra reading. In any case, you
should be comfortable working with groups, rings, ideals, and preferably
with modules. Field theory (including Galois theory) is not needed as such.
Algebraic geometry started off by studying solutions of certain polynomials,
which are called varieties. One then studies maps between
such objects, properties of such objects and of the maps between them, etc.
In more recent times, this theory was vastly generalized to associate
to collections of rings certain
geometric objects called schemes. Much of the theory of varieties
generalizes to schemes; however things get technically complicated.
The theory of schemes has become the accepted
common language in algberaic geometry these days. In this course, we
will study varieties and schemes almost in parallel, using varieties
to get the geometric intuition and schemes to get the algebraic finesse.
The goal is to learn
the language and some of the major results. Thus we will focus on breadth
rather than depth, skipping proofs for the most part.
The topics to be covered include varieties, schemes, morphisms,
basic properties and geometry of varieties and schemes, sheaf cohomology
and Serre duality (in particular the Riemann-Roch theorem). The sections
to be covered from the text are roughly I.1 to I.3, II.1 to II.8,
III.1 to III.7, and IV.1.
It is not clear
how much of this material we will cover. If needed some of it may get
postponed to a topics course in Spring 2009.
The grade will be based on homeworks and class attendance.
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