MTH426: Algebraic curves

[Cr:4, Lc:3, Tt:1, Lb:0]

Noetherian rings and Noetherian modules, Hilbert basis theorem, affine space and algebraic sets, ideal of a set of points, algebraic subsets of a plane, Hilbert's Nullstellensatz, integral and algebraic extensions.
Coordinate rings, affine coordinate transformations, discrete valuation rings, ideals with finitely many zeros, multiple points, tangent lines and local rings.
Projective space and projective algebraic sets, affine and projective varieties, product spaces, linear system of curves and Bézout's theorem, Max Noether's theorem.
The Zariski topology, morphism of varieties, algebraic function fields and dimension of varieties, rational maps.
Rational maps of curves, blowing up of a point in $\mathbb{A}^2$ and $\mathbb{P}^2$ , quadratic transformations and non singular model curves.
Divisors, Riemann's theorem and the genus of a non singular model curve, derivations and differentials, canonical divisors and the Riemann-Roch theorem.