Hi again. I recently completed the MIT Mathematics Directed Reading Program, which happened over January, during which I read the first few chapters of the infamous Algebraic Geometry by Robin Hartshorne. Schemes are a foundational concept in algebraic geometry, but they’re notoriously difficult to understand, so I thought I’d do my best to explain them in a way that would’ve been intelligible to me as a beginner.

First, some motivation. Algebraic geometry deals with studying the geometric properties of algebraic curves, such as \(y=x^2\), using algebra. One property of this curve is that it intersects lines \(y=k\) at 0, 1, or 2 points, depending on the value of \(k\). Also, when there is exactly 1 point of intersection, which happens at \(k=0\), it’s actually in some sense a “double intersection,” since the equation for the intersection reduces to \(x^2=0\), which has a double root at \(x=0\). How can we handle this information algebraically in a way that generalizes well?

To make our lives easier, we’re gonna consider \(y=x^2\) over \(\mathbb{C}\) instead of \(\mathbb{R}\), since \(\mathbb{C}\) is algebraically closed. Now, this curve always intersects lines \(y=k\) at 2 points whenever \(k \ne 0\).

In classical algebraic geometry, there’s this notion of an algebraic variety. In our example, the set of points satisfying \(y=x^2\) is a variety. We denote this \(V(y-x^2)\), since \(y-x^2\) vanishes at these points. The set of points in the intersection of \(y=x^2\) and \(y=k\) is also a variety, \(V(y-x^2, y-k)\).

Notice that the polynomials we use to define the variety aren’t unique. For \(V(y-x^2, y-k)\), we could have just as well written \(V(x^2-k, y-k)\). This is the concept of an ideal: if \(y-x^2=0\) and \(y-k=0\), then every polynomial in the ideal generated by \(y-x^2\) and \(y-k\) is \(0\), including \(x^2-k\) as well as \(y^2-x^2y\), for example. More formally, we talk about \(V(I)\) for an ideal \(I\) of the polynomial ring \(\mathbb{C}[x, y]\) (polynomials in \(x\) and \(y\) with complex coefficients).

So what about our variety \(V(y-x^2, y-k)\)? To find the number of intersections, we need to decompose this variety into irreducible components. We can do this by just writing

\[V(x^2-k, y-k) = V(x-\sqrt{k}, y-k) \cup V(x+\sqrt{k}, y-k)\]

This comes from the concept of an intersection of ideals (you can check that the two ideals on the right have intersection equal to the ideal on the left). But this only works if \(k \ne 0\). If we have \(k=0\) then the variety is actually irreducible.

That’s not the end of the story though, because notice that \(V(x^2, y) = V(x, y)\). We lose the extra information in the \(x^2\). Encoding this information requires introducing a new structure: schemes!

So far, we’ve been considering varieties, which are simple sets of points, but schemes take this a step further: they are ringed spaces, which essentially means that we add a structure of functions on the space. For example, we’ve been considering the plane \(\mathbb{C}^2\) this whole time. As a scheme, this is called \(\operatorname{Spec} \mathbb{C}[x, y]\), and the functions are polynomials in \(x\) and \(y\). The variety from before, \(V(y-x^2)\), is now called \(\operatorname{Spec} \mathbb{C}[x, y]/(y-x^2)\), and the functions are polynomials in \(x\), since in some sense the curve is one-dimensional. (Notice that, for example, the function \(y\) is identical to the function \(x^2\). What’s actually happening here is an “ideal quotient”: two polynomials are identical if their difference is in the ideal \((y-x^2)\).) Now, if we consider the variety \(V(x^2, y)\) from before, we get

\[\operatorname{Spec} \mathbb{C}[x, y]/(x^2, y).\]

On this scheme, the function \(x\) is not the same as \(0\), which is not true for \(\operatorname{Spec} \mathbb{C}[x, y]/(x, y)\). I won’t explain this fully, but the intuition is that \(y=x^2\) and \(y=0\) have the same tangent direction at \((0, 0)\), so the function \(x\) is sort of nontrivial on their intersection. This space \(\operatorname{Spec} \mathbb{C}[x, y]/(x^2, y)\) is called a “point with infinitesimal fuzz.”

After a lot of effort, we’ve managed to formalize the idea of double intersections. The point of the whole scheme idea is that it readily generalizes to more general situations, and even other rings such as \(\mathbb{R}\), \(\mathbb{Q}\), and \(\mathbb{Z}\).

Well that’s about it for now. I was originally going to include more examples and also talk about the correspondence between line bundles and divisors, which is what we actually presented on at the end of DRP. But since I ended up writing this entire thing assuming no knowledge of commutative algebra, I might as well put that in a separate post. This was at least informative in showing that it is actually possible to explain the intuition behind algebraic geometry without using commutative algebra. For one, the concept of an ideal arises very naturally.

Further reading:

  • Algebraic geometry on Wikipedia
  • Robin Hartshorne - Algebraic Geometry (read at your own risk)
  • Evan Chen - An Infinitely Large Napkin, parts XVIII and XIX. There’s a list of 18 examples of (affine) schemes in chapter 79 which is really useful; I actually don’t remember seeing this last time I read the book, so it must be pretty new (at least, within the last few years).