Algebraic sets, varieties, plane curves, morphisms and rational maps, resolution of singularities, Riemann-Roch theorem.
Algebraic Curves: An Introduction to Algebraic Geometry, Fulton, Addison Wesley, 1989.
MATH 791 and MATH 792.
The course of "Algebraic Curves" intends to cover the basic materials in algebraic curves from algebraic geometry point of view. The notion of algebraic curves is called Riemann surfaces in differential geometry. We will start from the basic notion of affine and projective varieties from algebraic geometry. Then we go to basic notion of algebraic curves. Our goal is to prove Riemann-Roch theorem for curves. If students are interested, and the time permitted, I will cover some fancy stuff about twisted curves (orbifold curves) and Riemann-Roch theorem for orbifold curves.
This course is a start up for Algebraic Geometry, and also for the research area on Gromov-Witten theory and Donaldson-Thomas theory in modern enumerative geometry.
(Jiang 2014 )