Mathematics 707 - Algebraic Geometry

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 an 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. Topics will include projective varieties, resolution of singularities, divisors and differential forms. Examples will include algebraic curves of low genus and surfaces in projective 3-space. 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. Prerequisites: Commutative algebra (rings and modules) as covered in 611-612. Some prior experience of manifolds would be useful (but not essential).

Open to Graduate students only. MATH 611 & 612