Elliptic curves. Algebraic curves, Riemann surfaces, and function fields. Singularities. Riemann-Roch theorem, Hurwitz's theorem, projective embeddings and the canonical curve. Zeta functions of curves over finite fields. Additional topics such as Jacobians or the Riemann hypothesis.

The theory of algebraic curves (aka Riemann Surfaces) is
the model for the development of algebraic geometry in general, and
plays a central role from number theory to physics. The core of the
theory is based in an range of examples and constructions for smooth
curves in projective space over the complex numbers. These are the
subjects of this course.

The course will begin with a rapid tour of the necessary definitions
and theorems in part 2 of the prerequisites to establish a common
language and background.

Then I'll follow the outline of the text, alternating basic
theoretical topics such as the Jacobian variety, the Hilbert scheme,
and the character of hyperplane sections with the treatment of
embeddings of curves of low genus in projective spaces.

Throughout, my goal will be to provide a feeling for the landscape of
algebraic curves and intuition about the examples that motivate the
field, rather than detailed proofs of difficult theorems.

Repeat Rules

Course is not repeatable for credit.