Overview
We will bring to life the abstract theory of schemes with the beautiful phenomena which appear in dimensions 1 and 2.
Topics (not necessarily exhaustive)
Riemann surfaces and the Riemann existence theorem
Divisors and the Riemann-Roch theorem
Curves of genus 0 (over a non-algebraically closed field)
Elliptic curves and their moduli
The Abel-Jacobi theorem
Moduli spaces of curves
Basic surfaces: Rational, Ruled, del Pezzo
The 27 lines on a cubic surface
Surface singularities
Elliptic surfaces: NĂ©ron-Severi and Mordell-Weil lattices
K3 surfaces
Expectations
If you are taking this class for credit, I expect you to attend class. By the end of the semester, I'd like you to write a short exposition (10-20 pages) of one of the topics we cover.