# Algebraic Geometry Part II

Welcome to the course webpage for Part II algebraic geometry, taught in the 2023 Lent term at Cambridge.

Instructor: Dhruv Ranganathan (E 1.01). Meeting time: MWF 12-1 PM (MR4).

Overview: The course will be an introduction to the basic theory of affine and projective algebraic varieties, as well as the theory of algebraic curves. These objects have been, and remain, the fundamental objects of interest in algebraic geometry. The material brings together ideas from algebra, topology, and geometry. We will try to learn the basic language, keep an eye on the richest examples of interest, all with a view towards modern developments.

Resources: I will provide TeX lecture notes at this link. The file will be updated roughly weekly, and will contain slightly more information and may frequently contain references and details that were not provided in lecture. There will also be four example sheets which will become available at the usual link.

There is no textbook that covers exactly the material that we will cover, and the TeX notes form the main resource. Texts are always good for different presentations however. In the first part of the course, texts of Hulek (Elementary Algebraic Geometry) or Reid (Undergraduate Algebraic Geometry) are excellent, as is Fulton (Algebraic Curves). The material on algebraic curves forms roughly the last third of the course, Fulton's text contains everything necessary, but Kirwan (Complex Algebraic Curves) is very insightful.

Schedule:

January 20: Introduction, basic questions, definitions of affine varieties.

January 23: Irreducible affine varieties and irreducible decompositions.

January 25: Ideals from varieties and the coordinate ring. Nullstellensatz. Morphisms.

January 27: (Mark Lectured) Morphisms and examples of morphisms. Rational functions.

January 30: The local ring and its maximal ideal. Introduction to projective space and projective geometry.

February 1: Projective varieties, homogeneous ideals, and basic examples. The Segre surface.

February 3: Closing up affine varieties to projective ones. Quadratic hypersurfaces. The projective Nullstellensatz.

February 6: Rational functions and rational maps on projective varieties.

February 8: The function field and the local ring. Relation between affine and projective function fields.

February 10: Rational maps and morphisms.

February 13: More on rational maps and morphisms. Birationality and tangent planes to hypersurfaces.

February 15: Tangent spaces to affine and projectie varieties. Linearizing rational maps and morphisms.

February 17: Linearizing morphisms. A few examples of theorems. Properties of the function field.

February 20: Generating the function field and birationality to hypersurfaces. Hilbert's Nullstellensatz.

February 22: Algebraic curves. Local structure of an algebraic curve.

February 24: Maps from smooth projective curves to projective space are regular. Morphisms between curves.

February 27: Degrees of morphisms, ramification degrees, and the basic structure of maps between curves.

February 29: More on morphisms between curves.

March 1: Divisor theory on curves, principal divisors, linear equivalence, and the Picard/Class group.

March 3: Divisors associated to hyperplane sections. Bezout's theorem. Basic inequalities for dim L(D).

March 6: Generalities on differentials, derivations, and calculus on algebraic curves.

March 8: The divisor associated to a rational differential.

March 10: Definition of the genus. The canonical divisor class of a plane curve.

March 13: Riemann-Roch and its consequences: the Riemann-Hurwitz formula and the group law on an elliptic curve.

March 15: Equations for abstract curves, hyperelliptic curves, and canonical embeddings.

Where to go next?

If you would like to go a little bit further in exploring algebraic geometry, I recommend trying to first gather up a good cluster of examples. Joe Harris's book "Algebraic Geometry: A First Course" is a great start. In order to solidify the technical foundations, the first chapter of Hartshorne's "Algebraic Geometry" should now be very accessible. The next significant new topic to study is the theory of schemes, which is the content of most of the above book by Hartshorne. If you're more geometrically inclined, you canm look at Donu Arapura's "Algebraic Geometry over the Complex Numbers".