Algebraic Geometry Part II

Welcome to the course webpage for Part II algebraic geometry, taught in the 2022 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.

Example sheets will appear here : 1 2 3 4.

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 21: Introduction, basic questions, definitions of affine varieties.

  • January 24: Order reversals and vanishing sets, irreducibility, Zariski topology.

  • January 26: Radical ideals and statement of the Nullstellensatz, morphisms.

  • January 28: Morphisms, rational maps, and local rings.

  • January 31: Projective space and projective varieties I.

  • February 2: Projective space and projective varieties II.

  • February 4: Eloise lectured on quadrics, the projective Nullstellensatz and the function field. Dhruv went to Paris.

  • February 7: Function theory for projective varieties, rational maps and examples.

  • February 9: Examples of rational maps and birational maps; implications for the function field. Segre and Veronese maps.

  • February 11: Function field and birationality. Tangent spaces and singular points on hypersurfaces.

  • February 14: Tangent spaces in general; the set of singular points is closed. Dimension.

  • February 16: Examples of theorems in the subject; birationality to hypersurfaces.

  • February 18: The Nullstellensatz and its proof. Algebraic curves and their basic structures.

  • February 21: The local structure of an algebraic curve.

  • February 23: Morphisms between curves and the structure theorem for morphisms of curves.

  • February 25: More on maps between curves; quasi-projective varieties and closed maps.

  • February 28: Divisors on curves; organizing the function field of a curve. Principal divisors and class groups.

  • March 2: Differentials on curves.

  • March 4: The divisor associated to a differential on a curve. The canonical class.

  • March 7: The genus of a curve. Differentials on plane curves. Riemann-Roch.

  • March 9: Applications of Riemann-Roch; Riemann-Hurwitz. The group law on the elliptic curve.

  • March 11: Embeddings of curves via Riemann-Roch. Hyperelliptic and canonical curves.

  • March 14: Geometry of lines in P3 and the Grassmannian.

  • March 16: The Grassmannian and the Hilbert scheme.

  • Our group picture from the last day is available to students in the class on request.