## Math 273, Spring 2009: Introduction to Algebraic Geometry

Instructor: Chad Schoen

Prerequisites: Math 252 (Commutative Algebra) and Math 272 (Riemann Surfaces)

### The subject:

Algebraic geometry is the study of the solution sets of systems of polynomial
equations. These objects are called algebraic varieties.
### Course Goals:

The basic objects of study in a first course on algebraic geometry are
quasi-projective varieties over algebraically closed fields. We will
investigate basic properties of these objects and learn some of the
tools applied to their study. Likely topics include:
Standard examples of varieties, dimension, regular maps, rational maps,
elimination theory, divisors, singularities,
coherent sheaves and their cohomology, cohomological invariants, rudiments of
intersection theory, a peek at algebraic surfaces.
The course treats concepts which are essential to further work
in algebraic geometry. To the extent that other areas of mathematics
require understanding of basic algebraic geometry concepts, this course
is important for students with a wide range of research interests.
Most researchers in the following fields need to be familiar with
the rudiments of algebraic geometry: number theory,
algebra, algebraic groups, quadratic forms, singularities,
complex analytic geometry, string theory.
Many topologists and differential geometers find knowledge of algebraic
geometry valuable in their work as well.

Research in algebraic geometry requires an understanding of
algebraic varieties over non-algebraically-closed fields and, more
generally, of schemes. Although these topics will not be treated in detail in
this course, the material will serve as preparation for these concepts.

### Text

The course will make use of the book, Algebraic Geometry, by Robin Hartshorne
as a source of reading and exercises.