Algebraic Geometry

Part II

Welcome to the course webpage for algebraic geometry, taught in the 2020 Lent term at Cambridge. This course is intended for Part II students in pure mathematics.

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

Overview: Algebraic geometry is an old and amazingly interdisciplinary and active subject, borrowing ideas from topology, differential geometry, number theory, and analysis. In this course the goal is to become acquainted with the basics, affine and projective varieties, and algebraic curves and some of the fundamental theory that governs their geometry.

Resources: We will not follow any textbook word-for-word, but excellent resources are the wonderful-and/or-irritating Undergraduate Algebraic Geometry by Reid and the first chapter in Hartshorne's Algebraic Geometry. I also like Algebraic Curves by Fulton which is free.

Lecture schedule:

  • January 17: Lecture cancelled due to Dhruv being unwell.
  • January 20: Mark lectured.
  • January 22: Mark lectured.
  • January 24: Lecture cancelled due to Dhruv being unwell.
  • January 27: Mark lectured.
  • January 29: Mark lectured.

Until this point, the material covered included definitions of affine varieties, the relationship between ideals and varieties, coordinate rings, irreducibility and irreducible decompositions, statement of Hilbert's Nullstellensatz.

  • January 31: Projective space, basic motivations, and homogeneous polynomials.
  • February 3: More on projective space, homogeneous ideals, projective varieties and their function fields.
  • February 5: Examples of projective varieties, plane curves and intersections, the twisted cubic.
  • February 7: The surface P1xP1. Homogenization for polynomials and ideals, computing projective closures.
  • February 10: Morphisms and rational maps. Birationality. Examples of morphisms.
  • February 12: Products of quasi-projective varieties. Blowups. Smoothness and tangent spaces for hypersurfaces.
  • February 14: Tangent spaces and dimension. Differentials and birational invariance and dimension.
  • February 17: Transcendence degree and birationality to hypersurfaces. A little more dimension theory.
  • February 19: Algebraic curves, smoothness, local parameters, orders of vanishing.
  • Residual: Nullstellensatz over uncountable fields, birationality and function fields, rationality of a cubic surface.

I hope to have time for: (1) A general degree 4 surface in P3 contains no lines. (2) A plane curve has finitely many bitangents lines and quartics have 28 bitangents (see picture above). (3) Hilbert polynomials and calculations for Veronese varieties.

Examples sheets: The example sheets will appear here.

Disclaimer: Algebraic geometry is a beautiful subject known for its technicality, but the reason for that technicality is that the range of geometric phenomena that appear in the subject is extremely large. Consequently, if you're serious about the subject, you should spend this course trying to see as much of this range as you possibly can -- you should explore and store examples. We'll see a very large number in lectures too of course. The examples in Harris's Algebraic Geometry: A First Course are wonderful and crucial, and I recommend learning a few examples a week just to build your geometric database.