Fall 2014

Classes: Tuesdays and Thursdays, 11:30AM--12:45PM in LGRT 1114.

Office hours: Tuesdays 3:00PM--4:00PM and Wednesdays 2:30PM--3:30PM in my office LGRT 1235H.

Commutative algebra (rings and modules) as covered in 611-612. Some prior experience of manifolds would be useful (but not essential).

Homeworks will be due every 1-2 weeks at the beginning of Thursday's class.

HW1 pdf. Due Thursday 9/25/14.

HW2 pdf. Due Thursday 10/9/14.

HW3 pdf. Due Thursday 11/6/14.

HW4 pdf. Due Thursday 12/4/14.

Algebraic geometry is the study of geometric spaces locally defined by polynomial equations. It is a central subject in mathematics with strong connections to differential geometry, number theory, and representation theory. This course will be a fast-paced introduction to the subject with a strong emphasis on examples. In the algebraic approach to the subject, local data is studied via the commutative algebra of quotients of polynomial rings in several variables. Passing from local to global data is delicate (as in complex analysis) and is either accomplished by working in projective space (corresponding to a graded polynomial ring) or by using sheaves and their cohomology. Topics will include projective varieties, singularities, differential forms, line bundles, and sheaf cohomology, including the Riemann--Roch theorem and Serre duality for algebraic curves. Examples will include projective space, the Grassmannian, the group law on an elliptic curve, blow-ups and resolutions of singularities, algebraic curves of low genus, and hypersurfaces in projective 3-space.

The red book of varieties and schemes, D. Mumford, googlebooks. Classic text.

Undergraduate algebraic geometry, M. Reid, googlebooks. Elementary introduction to algebraic geometry. Very accessible.

Basic algebraic geometry 1, I. Shafarevich, googlebooks. Fairly extensive introduction with few prerequisites.

Algebraic geometry: a first course, J. Harris, googlebooks. Recent book with lots of examples.

Algebraic geometry, R. Hartshorne, googlebooks. Standard text covering modern techniques in algebraic geometry. Rather intimidating for the beginner.

Principles of algebraic geometry, P. Griffiths and J. Harris, googlebooks. Describes the analytic approach to algebraic geometry. Full of instructive examples.

Hodge theory and complex algebraic geometry I, C. Voisin, googlebooks. Thorough development of the analytic approach (more careful than Griffiths-Harris, but fewer examples).

Undergraduate commutative algebra, M. Reid, googlebooks. An introductory text.

Introduction to commutative algebra, M. Atiyah and I. MacDonald, googlebooks. Classic text (very concise).

Commutative ring theory, H. Matsumura, googlebooks. Useful reference.

Commutative algebra with a view toward algebraic geometry, D. Eisenbud, googlebooks. Comprehensive text including the geometric point of view.