B3.3 Algebraic Curves (2022-23)

General Prerequisites:
Part A Topology. Introduction to Manifolds would be useful but not essential. Projective Geometry is recommended. Also, B3.2 (Geometry of Surfaces) is helpful, but not essential.
Course Term: Hilary
Course Lecture Information: 16 lectures
Course Weight: 1
Course Level: H
Assessment Type: Written Examination
Course Overview:
A real algebraic curve is a subset of the plane defined by a polynomial equation \(p(x,y)=0\). The intersection properties of a pair of curves are much better behaved if we extend this picture in two ways: the first is to use polynomials with complex coefficients, the second to extend the curve into the projective plane. In this course projective algebraic curves are studied, using ideas from algebra, from the geometry of surfaces and from complex analysis.
Learning Outcomes:
Students will know the concepts of projective space and curves in the projective plane. They will appreciate the notion of nonsingularity and know some basic features of intersection theory. They will view nonsingular algebraic curves as examples of Riemann surfaces, and be familiar with divisors, meromorphic functions and differentials.
Course Synopsis:
Projective spaces, homogeneous coordinates, projective transformations.
Algebraic curves in the complex projective plane. Irreducibility, singular and nonsingular points, tangent lines.
Bezout's Theorem (the proof will not be examined). Points of inflection, and normal form of a nonsingular cubic.
Nonsingular algebraic curves as Riemann surfaces. Meromorphic functions, divisors, linear equivalence. Differentials and canonical divisors. The group law on a nonsingular cubic.
The Riemann-Roch Theorem (the proof will not be examined). The geometric genus. Applications.