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.

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