Algebraic Geometry

Part II

Welcome to the course webpage for algebraic geometry, taught in the 2021 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 online.

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. Finally, I recommend looking at Gathmann's Algebraic Geometry Class Notes.

Lecture schedule:

  • January 22: Polynomials and their vanishing loci. Ideals. Hilbert's basis theorem. Affine varieties.

  • January 25: Properties of the vanishing locus under unions and intersections. Irreducibility and primality. The ideal construction and radical ideals.

  • January 27: The Zariski topology; statement of the Nullstellensatz. The coordinate ring and morphisms.

  • January 29: How to draw a plane curve of degree d. Rational functions, quasi affine varieties; local rings and their maximal ideals. The Zariski tangent space.

  • You should by now be able to complete Example Sheet I. You can find a glossary of the concepts we've defined here. At this stage, as long as you understand those concepts, have examples in mind for the properties and constructions we've discussed, and can engage with the questions on the sheet, you're doing fine.

  • February 1: Quick review of the basic concepts of affine varieties. Projective space.

  • February 3: Projective space, homogeneous coordinates, Zariski topology on projective space. Projective varieties.

  • February 5: Subvarieties of projective space: points, linear spaces, and quadric hypersurfaces. Homogeneous ideals.

  • February 8: Affine pieces of projective varieties, projective closure, homogenization of polynomials and ideals.

  • February 10: Rational functions on projective varieties; morphisms and local rings again.

  • February 12: Rational maps and morphisms on projective varieties. Basic notions surrounding rational maps.

  • In these attached notes, I explain two morphisms associated to a smooth plane conic: projection from a point on the conic and from a point not on the conic. There is an associated video on the Panopto thing.

  • February 15: Birational maps and birational equivalence. Products and the Segre embedding.

  • February 17: Dimension, tangent spaces, smoothness.

  • February 19: Birationality to hypersurfaces and transcendence bases. Krull's height theorem.

  • February 22: A few more remarks on dimension theory. Plane curves: basic geometry in low degree.

  • February 24: Basic geometry of abstract curves; smoothness and local parameters.

  • February 26: Morphisms from curves and their properties. The embedding question.