MATH 214A : Algebraic Geometry
- Course description: This
is the first of a two-quarter introduction to algebraic geometry.
This is an introduction to algebraic geometry,
the study of spaces defined by polynomial equations. Much of algebraic
geometry was motivated by Riemann's work on complex curves
in the 1850s, using topology
and complex analysis. The first half of the course studies
algebraic varieties over any algebraically closed field,
while building on the intuition from topology.
The second half of the course develops Grothendieck's
theory of schemes
from the 1960s, a broad generalization of the concept
of an algebraic variety. In particular, much of number theory
can be formulated in the language of schemes, but scheme theory
enriches even the special case of complex algebraic varieties.
Course outline: Affine and projective varieties over an
algebraically closed field. Irreducibility, dimension, regular functions,
rational functions. Local rings, singular and smooth points,
tangent spaces. Curves and their function fields. Morphisms of curves.
Intersections in projective space.
Sheaves and schemes.
Fiber products, morphisms of finite type, separated morphisms.
Proper morphisms, the Proj construction, finite morphisms.
Coherent and quasi-coherent sheaves.
Instructor: Burt Totaro.
E-mail: totaro@math.ucla.edu.
Lecture: MWF 2-2:50,
MS 7608.
Office hour: Monday, 3:00, MS 6136.
Textbook: Hartshorne's
Algebraic Geometry (Springer)
is the main book for the class. The UCLA Store should be selling it in their textbook department on floor A.
Roughly, I will cover Chapter I and sections
1-5 of Chapter II. Other useful
books include Kempf's Algebraic Varieties (Cambridge),
a short book that gets surprisingly far,
and Vakil's The Rising
Sea, free on the web.
Complex algebraic geometry (using differential geometry
and complex analysis) is not the main
focus of the class, but it is an important alternative point
of view. Huybrechts's Complex Geometry (Springer)
is a good introduction to that point of view.
Prerequisite: Math 215A
Commutative Algebra or permission of instructor. If you have not taken
Math 215A, please read Atiyah-Macdonald's Introduction to Commutative
Algebra (a short classic) in advance, and do some of the exercises.
Grading: Based on three homework sets.
Course web page: http://www.math.ucla.edu/~totaro/214a.1.20w/index.html
Homework 1 (due January 22, 2020).
Homework 2 (due February 19, 2020).
Homework 3 (due March 9, 2020).