Introduction to Algebraic Geometry (5320)

Instructor: Arend Bayer
Class time and room: TuTh 8am-9.15am.
Office: MSB 206

Office hours: Monday/Tuesday 2pm-3pm, Thursday 9.30 am - 10.30 am. Feel free to drop by any time, or email me to set a time to meet.
Phone:486-4237
E-mail: arend.bayer@uconn.edu

Homework:
Homework 1 (due 9/15)
Homework 2 (due 9/29)
Homework 3 (due 10/13)
Homework 4 (due 11/1)
Homework 5 (due 11/10)
Homework 6 (due 12/1)
Homework 7 (due ???)

Some Remarks and partial solutions to homework 3. Some suggested exercises about affine schemes.

Course content:
This course will be an introduction to the basic concepts of algebraic geometry. The main object of study will be affine and projective varieties, which are the solution sets of (homogeneous) polynomial equations. The main theme will be the interplay between algebraic properties (of rings of polynomials, and of the defining equations) and geometric properties.

A detailed list of anticipated topics is: Hilbert's Nullstellensatz, Zariski topology, regular functions, the Zariski tangent space, smoothness, degree, the Hilbert polynomial, Bezout's theorem, rational maps, divisors, line bundles and maps to projective spaces, Riemann-Roch theorem for curves. (This list is too long for the first semester, and I will choose among these later.)

We will illustrate these concepts with many examples, such as projective spaces, Grassmannians, quadrics, elliptic curves, Segre and Veronese embeddings, and blow-ups.

The prerequisite for this class is completion of the algebra sequence (5210, 5211). Introduction to Geometry and Topology (5310) is recommended, but it can be taken concurrently and exceptions are possible. I will introduce necessary background in commutative algebra as needed.

The main reference for this class will be lecture notes by Andreas Gathmann, available on his webpage.

Useful references:
Gathmann: Lecture notes. The main reference for this course.
Fulton: Algebraic Curves .
Hartshorne: "Algebraic Geometry". This is an exhaustive introduction to the techniques of modern algebraic geometry. Only chapter 1 will be relevant for this semester.
Shafarevich: "Basic Algebraic Geometry" (Volume 1).
Harris: "Algebraic Geometry - a first course". A great book, but the title is a bit misleading: it is full of interesting examples, many of which are quite challenging to understand completely.
Eisenbud, Harris: The geometry of schemes. This is great as additional reading on schemes.
Hulek: Elementary Algebraic Geometry. The first five chapters of this book are, to some extent, a more concrete version of Gathmann's sections 1-4.