As of late, I’ve been trying unsuccessfully to learn about Hilbert and Quot schemes. Nitin Nitsure’s notes (which appear in the book FGA Explained) are a good source, though they are also quite technical. I’ve been trying to read them, and the relevant parts of Mumford’s Lectures on curves on an algebraic surface. While doing so, I took a bunch of my own notes. They are right now unfinished, but they do cover the semicontinuity theorem and a small piece of the cohomology and base-change story. In addition, there is a little additional material (drafted) on applications of this to line bundles, which is more or less why Mumford develops all this in Abelian varieties.

I probably won’t get a chance to revise them further now. In fact, I’m going to take a long break from the “write lots of notes” mentality that has recently gripped me to focus on short, and ideally more-or-less self-contained posts in the future, not least because of limitations of time. This is something I’ve always had difficulty with: even my high school English teachers always had to tell me to shrink my essays. In a sense, though, the blog medium really is about pithy bites of profundity. Being short may take less time, but it’s also intellectually harder than just listing the main theorems in some small subsubsubfield and listing all the proofs in detail.

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