PMATH 764: Introduction to Algebraic Geometry
PMATH 764: Introduction to Algebraic Geometry
Lectures: MW 11:30-12:50 (MC 4063).
Office hours: T 10:00--12:00 and Th 15:00--17:00.
Course information:
Outline.
Course web page:
https//:learn.uwaterloo.ca.
Course description:
An introduction to algebraic geometry through the theory of algebraic curves. General Algebraic Geometry: affine and projective algebraic sets, Hilbert's Nullstellensatz,
co-ordinate rings, polynomial maps, rational functions and local rings. Algebraic Curves: affine and projective plane curves, tangency and multiplicity, intersection
numbers, Bezout's theorem and divisor class groups.
Outline of topics:
- General algebraic geometry:
-
Affine and projective varieties.
-
Coordinate rings, fields of functions, and local rings.
-
Polynomial and rational maps.
-
Dimension and smoothness.
-
Algebraic curves:
-
Affine and projective plane curves, and their local properties.
-
Intersection numbers and Bézout's Theorem.
-
Divisor class groups.
-
Elliptic curves and the group law on the cubic.
Prerequisites:
PMATH 345 -- Polynomials, rings and finite fields or
PMATH 347 -- Groups and Rings (or the equivalent).
Corequisite:
PMATH
348 -- Fields and Galois Theory (or an equivalent course on field
theory).
References:
-
Algebraic curves: an introduction to algebraic geometry, by W.
Fulton;
-
Algebraic geometry, by R. Hartshorne;
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Algebraic curves and Riemann surfaces, by R. Miranda;
-
Algebraic geometry: a first course, by J. Harris;
-
The red book of varieties and schemes, by D. Mumford;
-
Basic algebraic geometry, by I. R. Shafarevich.
Lecture Notes:
Week 1.
Week 2.
Week 3.
Zariski's proof of the Hilbert
Nullstellensatz.
Week 4.
Assignments:
Assignment 1.
Assignment 2.