M4P33 Algebraic Geometry

2016-7, Term 2

Lecture notes (all lectures, including mastery material in Appendix A)

Problem sheets


Lecturer: Martin Orr

Assessment: Two courseworks (5% each), one exam (90%)
Coursework deadlines: 16 Feb, 14 Mar


What is algebraic geometry?

Algebraic geometry is the study of shapes defined as solutions to systems of polynomial equations in more than one variable. We call these shapes algebraic varieties. Examples include:

Image of circle Image of cardioid Image of quadric surface
The circle x^2 + y^2 = 1 Cardioid (x^2 + y^2 - x)^2 = x^2 + y^2 A quadric surface x^2 + y^2 = z^2 + 1

The types of questions we ask in algebraic geometry include:

Why study algebraic geometry?

The beauty of algebraic geometry lies in its combination of geometric ideas and algebraic tools. Algebra is used to give precise definitions and proofs of the foundations, in place of the analysis you may be used to from other kinds of geometry. A key benefit is that this allows us to use geometric ideas to study solutions of polynomials over any field, not just the real or complex numbers.

Areas of mathematics which use algebraic geometry include number theory, representation theory, coding and cyptography and string theory. It also has applications in mathematical biology, algebraic statistics and control theory, where interesting problems can be described in terms of the solutions to some system of polynomial equations and then studied using the methods of algebraic geometry.

What will be in this course?

In this course, we will aim to focus on geometric intuition and examples. We will make use of commutative algebra, especially calculations with polynomials, but aim to avoid more advanced or technical concepts from algebra.

We will see the interplay between algebra and geometry in the foundations of the subject, for example the Nullstellensatz and basic properties of dimension. The core concepts will be illustrated with many interesting examples. We will not just give formal definitions of properties such as dimension and smoothness, but also see ways of calculating these for concrete examples.

By the end of this course:

A cubic surface with the 27 lines on it

How does this course relate to other courses?

What books are useful for this course?

Attributions for figures

The figures of the circle and cardioid were created by Martin Orr using gnuplot. These figures are in the public domain.

The figure of a quadric surface is a cropped version of a figure created by Sam Derbyshire and obtained from Wikimedia Commons. It is used under the CC BY-SA 4.0 license.

The figure of a cubic surface with 27 lines was created by O. Labs and D. van Straten and obtained from the Cubic Surfaces Homepage at the University of Mainz.