Two definitions of a variety

This semester I am participating in a reading group that is reading Fulton’s textbook on Toric Varieties. My motivation is to understand recent developments in matroid theory better, but in the process I am finding a need to shore up some basic geometry knowledge. Abstractly I know there are two definitions of an algebraic variety but I couldn’t remember the “coordinate free” one. My mathematical sister Sasha Pevzner showed me a way to remember it and why it makes sense in the context of the first one. This post is all about that.

The first definition

The zeroth “definition” of a variety is as a curve we want to study geometrically. To make this more precise, we can state the first definition of a variety. This is the one that I have internalized from reading Cox, Little, and O’Shea’s book Ideals, Varieties, and Algorithms, is as “the set of solutions of a polynomial equation.” Lets think about how we can think of the parabola \(y = x^2 - 1\) as an algebraic object in order to study it geometrically.

I’ll regard this as a curve in the real plane. Since we are working in the real numbers, our polynomials should have real coefficients, and since we are working in the plane, we should have an \(x\) coordinate and a \(y\) coordinate. Thus our polynomials are in the polynomial ring \(S = \mathbb{R}[x,y]\). It is a common technique in solving linear differential equations to first solve a homogeneous version of the equation. A rough reason for this is that it is easier to reason about zero than other numbers. We’ll take the same perspective here. Instead of thinking about the equation \(y = x^2 - 1 \) we can think about the polynomial \(f = y - x^2 + 1\) with the understanding that \(y - x^2 + 1 = 0\). So then the variety corresponding to \(f\) contains the points \((0,-1)\), \((\pm 1,0)\), and \((\pm 2,3)\) among others. Lets introduce notation for a variety corresponding to \(f\): \[ V((f)) = \{(x,y) : y - x^2 + 1\}. \] The “data type” of \(V((f))\) is a set of points in the plane.

It is a fact that there is a correspondence between ideals and varieties. This basically comes down to the fact that if \(f = 0\) then \(gf = 0\) for any polynomial \(g\). So if \(g\) is any polynomial in the ideal \(I = (f)\subseteq S\) generated by \(f\), and \((x,y)\) is any point in \(V((f))\), then \((x,y)\) is a solution to \(g = 0\).

We can go backwards too, but it is a little more subtle. If \(g \in S\) is a polynomial that factors as \(g = pq\), then either \(p\) or \(q\) can be zero to solve \(g\). For example, consider \(g = (x+1)^2\). Then \(V((g)) = {(-1,y) : y \in \mathbb{R}}\) but this is the same as \(V((x+1))\). The two different ideals define the same variety. In order to “reduce” the ideal, we take the radical of the ideal. This is sort of like taking \(n\)th roots in an ideal so we denote the radical of an ideal \(\sqrt{J}\).

The second definition

Now we can introduce the second, to me more sophisticated but also more opaque, definition. Denote by \( \operatorname{Spec}(s) \) the set of all prime ideals of \(S\). One nice fact about prime ideals \(p\) is that \(\sqrt{p} = p\). Additionally, every maximal ideal is prime. The more sophisticated definition is that the variety of \(I\) is \[ V(I) = \{ p \in \operatorname{Spec}(s) : I \supseteq p \}.\] The “data type” of \(V(I)\) is a collection of prime ideals.

Points and maximal ideals

In algebraic geometry, one defines a “point” to be a maximal ideal. This makes sense when working over the complex numbers, because maximal ideals of \(\mathbb{C}[z_1, z_2, \ldots, z_n]\) all look like \((z_1 - a_1, z_2 - a_2, \ldots, z_n - a_n)\). Lets consider the zero set of this maximal ideal inside of \(\mathbb{C}^n\). (We want to think about \(\mathbb{C}^n\) because we have coefficients in \(\mathbb{C}\) and \(n\) many indeterminates.) We need every polynomial in the ideal to vanish on the whole zero set. Since \(z_1 - a_1\) must vanish, any point in the variety must have \(z_1\)-coordinate equal to \(a_1\). Since \(z_2 - a_2\) must vanish, any point in the variety must have \(z_2\)-coordinate equal to \(a_2\). Since \(z_3 - a_3\) must vanish, any point in the variety must have \(z_3\)-coordinate equal to \(a_3\). Since \(z_4 - a_4\) must vanish… you get the point. We have \(n\) linear equations and \(n\) unknowns, so if a solution exists it is a single point. The solution does exist - it is \((a_1, a_2, \ldots, a_n) \in \mathbb{C}^n\).

In \(\mathbb{R}[x,y]\), not every maximal ideal defines a point in euclidean space (for example \((x^2+1, y)\) doesn’t). On the other hand, any point in euclidean space does define a maximal ideal - \((x_0, y_0)\) defines the maximal ideal \((x - x_0, y- y_0)\). So defining a “point” to be a maximal ideal generalizes what we intuitively consider to be a point. The point (get it) of all of this is that it makes sense to call maximal ideals points. Then points in a variety are the maximal ideals which contain a specific ideal.

Points and varieties

In the first definition we considered points to be in a variety if they solved a polynomial equation. If we pick the second definition, we considered points to be maximal ideals containing a specific ideal. It would be pretty disheartening if these two definitions of varieties corresponded to completely different notions. Fortunately it works out. Lets see how with our example of the parabola works out.

We want the points \((0,-1)\) and \((1,0)\) and \((2,3)\) to be on the parabola \(f = y - x^2 + 1\). Certainly they satisfy the equation. So if we take \(I\) to be the ideal generated by \(f\) we would hope that the points are in the variety no matter which way define it. That means that we would hope that the maximal ideals \[m_1 = (x,y+1),\; m_2 = (x-1, y),\; m_3 = (x-2, y-3)\] contain the ideal \(I\). It is enough to show that \(m_1, m_2, m_3\) contain the polynomial \(f\) because \(f\) generates \(I\). Let’s write \(f\) in a bunch of different ways. We have \[f = x(-x) + (y+1),\] \[f = (x-1)\cdot (-x-1) + y,\] \[f = (x-2)\cdot (-x-2) + (y-3).\] The first way shows that \((x, y+1) \subseteq (y-x^2+1)\). The second way shows that \((x-1, y) \subseteq (y-x^2+1)\). The third way shows that \((x-2, y-3) \subseteq (y-x^2+1)\).

So whether we regard these points as given in the Cartesian plane by \((x,y)\) coordinates, or as a maximal ideal of \(\mathbb{R}[x,y]\), they are in the variety defined (using definition 1) by the polynomial \(f = y - x^2 +1\) or (using definition 2) by the ideal \( (f) \subseteq \mathbb{R}[x,y]\).