Now that we can attend seminars all over the world, beginning algebraic geometers may be encountering the Hilbert scheme everywhere. At first glance, however, the idea of the Hilbert scheme is so capacious that it can be hard to grasp.
So, in this post, I want to sketch a path that an interested reader (or student seminar) could follow in beginning to understand the Hilbert scheme. (Some of the links may require your library to have access.) The topic came to mind because of my current joint work with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, and Maria Yakerson, “The Hilbert scheme of infinite affine space.”
First, the general definition: for a projective variety (or scheme) X over a field, the Hilbert scheme of X classifies all closed subschemes of X. The existence of the Hilbert scheme (as a union of projective schemes) reflects a basic feature of algebraic geometry: families of algebraic varieties are parametrized by an algebraic variety (or scheme). But then we have to find ways to analyze the Hilbert scheme in cases of interest.
To start the path of reading, there is an excellent introduction by James McKernan, a 10-page set of MIT lecture notes. It discusses a special case, the “Hilbert scheme of points”, with several examples.
The general construction of Hilbert schemes, due to Grothendieck, is outlined in several places. Perhaps the easiest to read is chapter 1 of Harris and Morrison’s book Moduli of Curves. The most technical step of the proof is not included there (roughly, the fact that there is a uniform bound for the equations of all subschemes of projective space with a given Hilbert polynomial). It might be reasonable for learners to come back to this step later. A standard reference for this step is chapter 14 of Mumford’s book Lectures on Curves on an Algebraic Surface.
Although the Hilbert scheme is hard to understand in full detail, there is a clear — computable! — description of its Zariski tangent space at any point, even where it is singular. Namely, if S is a closed subscheme of a projective scheme X over a field, then the tangent space to Hilb(X) at the point [S] is H^0(S, N_{S/X}), the space of global sections of the normal sheaf. Computing this group in examples is essential for getting to grips with the Hilbert scheme. As a start, you can look in chapter 1 of Harris-Morrison or chapter 1 of Kollár’s book Rational Curves on Algebraic Varieties.
Finally, for many applications, it is important to go one step further and understand the “obstruction space” as well as the tangent space for the Hilbert scheme. This is a great setting in which to learn deformation theory. Roughly, the obstruction space tells you the number of equations needed to define the Hilbert scheme near a given point. There are many possible introductions to deformation theory; let me recommend Sernesi’s book Deformations of Algebraic Schemes. Section 3.2 addresses the Hilbert scheme, with examples and exercises.
There is a vast literature on Hilbert schemes in particular settings, such as the Hilbert scheme of points on a surface. But I hope what I’ve said is enough for you to start exploring.
Image by @kernpanik; license CC BY-NC-SA 4.0.
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