What algebraic geometry is about

A good introduction by Jack Huizenga, from quora.com

At a very bare-bones level, algebraic geometry is technically the study of solutions of systems of polynomial equations. As it is actually studied and practiced, however, it attempts to answer qualitative and geometric questions about such solution sets.

In high school algebra, you might do things like try to determine the precise solutions to a system of equations such as

y = x

x^2+y^2=2

Here this is the intersection of a line and a circle, and consists of two points. It is possible to determine precisely the coordinates of those two points, and there isn’t a whole lot of “geometry” going on.

But what if we consider a system of the form

p(x,y,z)=0
q(x,y,z)=0

where p,q are two polynomials in three variables? We’d expect that each equation describes a surface in 3-space, and that their intersection is a curve.

What does it mean to “solve” a system of this form? Perhaps it means to fully describe the solution set in some kind of explicit way. For instance, one could try to parameterize the curve of intersection. That is, maybe you can write down some function

f: \mathbb{R}  \rightarrow \mathbb{R}^3

such that

p(f(t))=0

for all t, and likewise with the other equation.

For example, maybe we’re considering the system

z=0

x^{2} + y^{2} - z^{2} = 1

Then the solution set can be parameterized by

f(t)=(\cos t, \sin t,0)

Parameterizing the solution set could be viewed as the algebraic way of describing a zero set of a system of polynomials. We learn a lot (in some sense everything) about the individual solutions, but the structure of the set of solutions as a whole is not really illuminated.

On the other hand, in algebraic geometry one usually doesn’t care so much about explicit equations. In the last example, you might just say the intersection is a circle and be done with it. Typically the types of questions you ask in algebraic geometry are more qualitative questions about the geometric structure of the solutions. Of course, the theory is capable of asking precise questions about, for instance, the coordinates of points in the solution set, but it usually doesn’t have too much to say. The interesting things you can learn from the theory involve deeper information about the whole collection of solutions, instead of individual solutions.

The cubic surface. For a less trivial example, a cubic surface is a surface in 3-space defined by a polynomial equation of degree 3. One such example is the Fermat cubic surface

x^3+y^3+z^3=1

The Fermat cubic surface.

If we are given two different cubic surfaces, each given by their own polynomial of degree 3, it is not so clear from an algebraic perspective what intrinsic properties the two surfaces have in common. They are just given by some equation, and it is hard to directly make analogies between them besides the fact that they have somewhat similar equations.

In algebraic geometry, you would study the cubic surface not necessarily by looking for instance at the coordinates of its points, but by trying to answer questions such as

What kinds of curves are on this surface?
Are there other ways of describing this surface in terms of some fundamental geometric operations?
Is it possible to parameterize this surface by a plane? (Without necessarily caring about what the parameterization is in terms of explicit formulas.)

Let me try and describe a bit about each of these three questions in the case of a cubic surface. Let’s assume our surface is smooth, which is a technical condition ensuring the surface has no singularities. In other words, if you zoom in really close at a point of the surface, it looks basically like a plane.

Curves on a cubic surface. There are obviously many different curves which are contained in a cubic surface. For instance, if you intersect the surface with a plane, you’ll get a cubic curve in that plane. Most of the time, this looks something like this:

The typical plane section of a cubic surface.

If you were to intersect your cubic with some higher degree surface, such as a surface defined by an equation of degree 2, you’d get some complicated space curves.

It’s also possible to choose a plane which intersects the surface in such a way that the intersection looks like this:

Picture shamelessly copied from earlier.

In particular, the surface actually contains some lines. It is a fact that any smooth cubic surface has 27 lines. (In the earlier picture of the Fermat cubic, several of the lines are relatively easy to see. Some of the lines may be “complex” or “at infinity” and not visible in the finite real picture.) The lines on a cubic surface intersect in a beautiful pattern called the double-six configuration.

The double-six configuration of lines on a cubic surface. (Credit: Wikipedia)

Another description of the cubic surface. Consider the plane. There is an operation called blowing-up, which takes a point in the plane and replaces it by a line. This is perhaps best illustrated by the prototypical picture:

Schematic diagram of the blow-up of a plane.

Lines which pass through the point with different slopes are “stretched out” into a spiral staircase. Algebraic geometry is not so much the study of things cut out by specific polynomial equations as it is the study of things which can be described by polynomial equations, and this wonky staircase-thing can in fact be described by polynomial equations.

(Aside: if you are asked by security at an airport what your job is while trying to get on a flight, don’t say “I blow up planes.” Even if it’s true.)

It turns out, if you start from the plane and blow up six different points then the thing you end up with “is” a cubic surface. I put “is” in quotes here because the blowup here is not embedded as a surface in space; what this means is that there exists some polynomial map which embeds the blowup in space as a cubic surface.

Conversely, given any smooth cubic surface, there is a collection of six points in the plane such that the surface is the blowup of the plane at those six points. The six lines introduced in the blowup are six of the 27 lines on the cubic surface.

Parameterizing the cubic surface by a plane. In fact, it is impossible to parameterize the cubic surface by a plane in the strictest sense: there is no way to make the points of the cubic surface correspond to the points of the plane in a natural one-to-one fashion. But the alternate description of the cubic surface in the previous subsection gives a natural way of parameterizing almost all of the points by the plane. In fact, the blowup of the plane at six points is basically just the plane with six points replaced by lines. Then away from these six lines, the cubic surface is parameterized by the plane with the six points thrown out. We say that the cubic surface is birational to the plane, and it is rational.

Closing remarks. As it is actually studied, algebraic geometry has a lot more to do with the geometric pictures I’ve presented here than it does the solving of polynomial equations from high school type algebra. To study higher-dimensional zero sets of systems of equations it is much more profitable to ask qualitative geometric questions than it is to ask about explicit equations and parameterizations.