Math 273, Fall 2005: Introduction to Algebraic Geometry

Instructor: Chad Schoen

Prerequisites: Math 252 (Commutative Algebra) and Math 272 (Riemann Surfaces)
It may be possible to take math 272 concurrently. Math 272 supplies useful intuition.
It also supplies technical knowledge about sheaves and their cohomology. Math 252 is the more important
prerequisite.

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. Topics: Quasi-projective varieties with numerous examples, regular maps, rational maps, elimination theory, singularities, divisors and maps to projective space. To the extent that time permits we will treat 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, complex differential geometry.

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 this course, the course will prepare students to study these concepts.

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Algebraic Geometry, by Robin Hartshorne