What is a variety?
Of course, I am not talking about the English word “variety” here, which everyone knows the meaning. I am referring to algebraic varieties. These are the primary objects one studies in algebraic geometry. I am no expert on the subject, but I have a little understanding of the concept. I would like to share my thoughts and insights by showing some elementary and common examples everyone would have seen sometime or the other - completely unaware that they are varieties in disguise!
§ Variety - Definition
Definition: A variety is a set of solutions to a system of polynomial equations, usually over the real or complex numbers.
An expert algebraic geometer might scorn my simplifications to the definition, but I think it captures the essence while still being formally accurate. There are also generalizations of this, but I won’t be discussing those here.
So simply speaking, say you have a set of polynomial equations (in one or more variables). If you compute the set of values that satisfy all these equations, you have a variety. This is a pretty simple definition. Before reading the following sections, I urge you to think of examples from the math you already know.
§ Some cooked up examples
One can easily just pick any set of their favourite polynomials, and solve them to make a variety. Let us try that.
Consider the following system of two polynomials: \(P = \{ x^2 - 2x + 1, x^2 - 3x + 2 \}\). This produces the variety \(Z(P) = \{ 1 \}\). Just a single element.
We can also consider polynomials with more common roots: \(Q = \{ (x-1)(x-2)(x-3), (x-2)(x-3)(x-4), (x-1)(x-3)(x-4) \}\). This produces the variety \(Z(Q) = \{ 2, 3 \}\).
§ Wait, where is the geometry in this?
Now one might naturally wonder, how does this connect algebra and geometry? And you already know the answer! When you plot the polynomials on a graph, you get a geometric object, some curve. Studying the roots of the polynomial is the same as studying the object’s intersection with the $X$ axis.
We can now expand to larger examples - multivariate polynomials. What is the variety given by this system: \(C = \{ x^2 + y^2 - 1 \}\)? Yes, it is the unit circle around the origin! So a circle is just an algebraic variety! So is a line or an ellipse or a hyperbola! Any geometric object which can be described by a polynomial equation in one or more variables is just a variety. And what is the system that generates this? The singleton set of polynomials that contains only its equation.
These are very simple examples, and just a glimpse to the field of algebraic geometry - the study of algebraic objects using geometry. A friend had once told me the formal definition of a variety (the one I wrote above), to explain something else that depended on this. And much later I made this connection randomly - that a circle or a line are just varieties. Though it might seem simple, I found it quite fascinating.
What other random geometric objects have you seen in your daily life that are varieties?
§ Further reading
If you find this subject interesting, here are a few things you could read/learn from:
- Cox, O’Shea, Little. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra
- Algebraic Geometry in the Time of COVID - Lecture series by Ravi Vakil
