Welcome to MATH 232A: Introduction to Algebraic Geometry I

Description: Introduction to complex algebraic curves, surfaces, and varieties.
Meeting Time: MW 10:30am - 11:45am @ SC 411

Major reference:

Note by Andreas Gathmann

Other reference:

  • Introduction to Commutative Algebra by M. F. Atiyah and I. G. MacDonald
  • Algebraic Geometry - A First Course by Joe Harris
  • Basic Algebraic Geometry 1: Varieties in Projective Space by Igor R. Shafarevich
  • Algebraic Geometry by Robin Hartshorne
  • Algebraic Geometry over the Complex Numbers by Donu Arapura

Contact information:

Man Wai (Mandy) Cheung @ Science Center room 505H
Office hour: Wed 1:30-2:30pm; Fri 9-10am
Send me an email to schedule a chat if you cannot make these office hours.

Homework: There will be a homework assignment biweekly. Late homework is accepted only with my approval.
  • Problem set 1 (due on Sep 18)
  • Problem Set 2: Harris: Exercise 1.19 + Hartshorne: I: 2.6, 2.9, 3.1, 3.2, 3.3 (due on Oct 2)
  • Problem Set 3: Harris: Exercise 2.12, 2.24, 6.8 + Hartshorne: I: 2.12, 2.15, 3.5, 3.7, 3.17 (due on Oct 16)
  • Problem Set 4: Harris: Exercise 7.12 + Hartshorne: I: 4.3, 4.4, 4.5, 4.6, 4.7 (due on Oct 30)
  • Problem Set 5: Harris: Exercise 11.6, 11.19, 13.1 + Hartshorne: 5.12 + Gathmann Ex 9.22, 12.4, 12.15 (due on Dec 2, no late!)

Grading: The final grade in the course will be based on the homework grades and a final project. The final project would be take-home writing project/ exam / presentation.

Schedule of the class:

Sep 4 -- Affine varieties
Sep 9 -- Affine varieties
Sep 11 -- Projective varieties
Sep 16 -- Examples
Sep 18 -- Regular functions, localization
Sep 23 -- Sheaf and Morphisms (Affine)
Sep 25 -- Morphisms, Product
Sep 30 -- Product, seperatedness, affine varieties
Oct 2 -- Regular functions on projective space, Segre embedding
Oct 7 -- Veronese Map, complete, Grassmannians
Oct 9 -- Plucker embedding
Oct 14 -- School holiday
Oct 16 -- Incidence correspondence, Rational maps
Oct 21 -- Birational maps, degree of rational maps
Oct 23 -- blow up
Oct 28 -- Dimension
Oct 30 -- continue dimension, tangent cone
Nov 4 -- Tangent space, Smoothness
Nov 6 -- Hilbert functions
Nov 11 -- Hilbert polynomial, Riemann-Roch for curves
Nov 13 -- Bezout's theorem
Nov 18 -- Kaufmann: Strassen's degree bound for arithmetic circuits
Nov 20 -- Cancelled with note: Bertini theorem
Nov 25 -- Boretsky: Grassmannian
Dec 2 -- Vo: linear system of conics in P^2