Autumn/Winter 2001-2002, MWF 9:30-10:20
Plane curves: Conics and cubics. The group law on cubics and applications. Higher-degree curves and their genus.
Affine varieties: Noetherian rings, Hilbert Basis Theorem, Zariski topology, Hilbert Nullstellensatz. Coordinate rings, morphisms, rational maps, dimension.
Projective Varieties: Examples, relation with affine varieties, birational equivalence. The tangent space from geometric and algebraic points of view. Singularities, resolutions. Quadric and cubic surfaces, the 27 lines on a cubic surface.
MATH 508, Winter Quarter
Algebraic Curves: local properties of plane curves, projective plane curves, Max Noether's Theorem and applications. Resolution of
singularities of curves. Divisors, derivations and differentials, the canonical divisor, the Riemann-Roch Theorem. Examples. Time permitting: cohomology and Serre duality.
BOOKS:
Miles Reid, Undergraduate Algebraic Geometry William Fulton, Algebraic Curves