MATH 631: ALGEBRAIC GEOMETRY I

Professor Karen E. Smith

Algebraic geometry is one of the most highly developed and beautiful branches of mathematics. It is also one of the oldest and most central, interacting with and influencing the fields of commutative algebra, non-commutative algebra, representation theory, algebraic number theory and complex analytic geometry in fundamental ways. A working knowledge of the language and ideas of algebraic geometry are also useful to mathematicians working in an even more broad range of topics including algebraic combinatorics, topology, and differential geometry. Some branches of modern physics and applied computer science also draw heavily from algebraic geometry.

This course will be a basic and broad introduction to Algebraic Geometry, for students at the ``second year'' level and higher. Topics covered will include affine and projective varieties, Hilbert's Nullstellensatz, the Zariski topology, the sheaf of regular functions, regular and rational maps, dimension, the Zariski tangent space, the concept of smoothness, degree, the Hilbert polynomial, blowing up, divisors, line bundles and maps to projective space, the Riemann-Roch formula for curves. Considerable attention will be paid to the rich examples of classical algebraic geometry: Grassmannians, flag varieties, curves, Segre and Veronese maps, blow-ups, quadrics, determinantal varieties. We will also touch on the idea of schemes and of varieties as functors, though these concepts will be developed only as examples, not formally or precisely.

The prerequisite for Math 631 is mathematical sophistication at the level of someone who has completed all the alpha courses. Ideally, students have seen Math 593/4, Math 596, Math 591, as well as much of the material from Math 614 (Commutative algebra). For beginning students who haven't seen Math 614, spending time this summer doing exercises from Atiyah MacDonald is highly recommended. However, Math 631 will be self contained, in the sense that many of the results we will use from commutative algebra can be "black-boxed."

Math 631 is a "seriously graded class." Students are expected to attend and participate in class, to complete weekly problem sets, plus write a final paper. The workload is not for the faint-of-heart. However, students who are already candidates and would like to enroll can come talk to me about an appropriate method of evaluation.

The closest thing to a text is Shafarevich's Basic Algebraic Geometry'' (Volume I) , which is available in the paperback ``Springer Study Edition'' series. We'll also draw from the professor's own An Invitation to Algebraic Geometry (second edition) and Harris's Algebraic Geometry: A first course, which contains lots of good examples. Other classics in the field include Hartshorne's Algebraic Geometry, which is a good reference book and Eisenbud's Commutative Algebra with a View towards Algebraic Geometry, which might be helpful for those who want a more algebraic point of view.