In order to keep this blog alive and well, I thought I would write a quick post on a topic I have been thinking about lately to prepare for a seminar talk.
“Points in the plane” seem to be quite harmless: we want to study properties of the possible configurations of finitely many points. The properties we might be interested in include:
- How many points are there? Is this number even or odd?
- How many of the points lie on the same line? (A line is given by an equation of degree 1)
- Similarly, we could ask how many of the points lie on an equation of degree 2, 3, etc.
A good tool to answer such questions is Commutative Algebra. We turn a questions about geometry (where the points are) into a question about equations (solving for “x”).
Consider some polynomial function that is defined on the plane. For each location it takes some value. For example, we could have .
We are interested in functions that vanish at each of our points. If is our arrangement of points, then we are interested in functions that vanish at every point in or, we say, that vanish “on “. The collection of all functions that vanish on forms an algebraic structure called an ideal. We call this ideal .
Although at first it seems a round-about way of doing things, this algebraic structure is much neater to work with than the original configuration of points, and allows us to obtain succinct descriptions of our collection of points.
The ideal is formed of particular “building block” functions from which the other functions are made: these are called the generators of the ideal. For example, it may be that all functions in can be build from the two functions and . This first equation, , is linear (degree 1) and the second equation, , is quadratic (degree 2).
For a collection of 8 points in the plane, we can show that the ideal is generated by two equations of degree 3 (cubics) and 1 equation of degree 4 (quartic). Here is a picture of one such configuration of 8 points, taken from the book “The Geometry of Syzygies” by David Eisenbud:
The points in our configuration are represented by black dots. All three of our functions vanish at each of the dots – there are always three lines passing through each dot, one line from each function.
For a different arrangement of black dots, we would have a different arrangement of the two cubic equations and one quartic, so that each of the new lines would pass through each of the new points.