Commutative Algebra and Introduction to Algebraic Geometry

Topics

  • Hilbert’s basis theorem,
  • Hilbert‘s Nullstellensatz,
  • (quasi-)affine and (quasi-)projective varieties,
  • sheaves and ringed spaces
  • dimension,
  • Zariski topology,
  • local rings,
  • morphisms and rational maps,
  • blow-ups smoothness,
  • intersection multiplicities, Bézout‘s theorem,
  • plane curves and resolution of singularities

The plan is to follow the lecture notes of Andreas Gathmann and add some commutative algebra where needed.

Literature

  • Atiyah McDonald, Introduction to commutative Algebra,
  • R. Hartshorne, Algebraic Geometry, Springer.
  • K. Hulek Elementary Algebraic Geometry,
  • M. Reid, Undergraduate Algebraic Geometry
  • Cox, David A.; Little, John and O'Shea, Donal, Ideals, varieties, and algorithms

Links to the books are found here.

Prerequisites
- Linear Algebra I + II
- Algebra will be useful
 
Dates
Lecture 1: Mondays 12:15 - 13:45, SR6 Building E2.4
Lecture 2: Wednesdays 14:15 - 15:45, SR6 Building E2.4
 
Tutorial, Mondays 14:15-15:45 Zeichensaal U.39 Building E2.5; (Except on May 13th, where we will be in SR 9 instead.)
Office hours: by appointment
 
Exercises
There will be weekly exercise sheets. The solutions are discussed in the tutorial.
You are encouraged to solve the exercises yourself. For feedback on your solutions, you can hand them in at the end of the lecture.
  1. Sheet
  2. Sheet
  3. Sheet
Exam
There will be an oral exam.