I’d like to use this post to try to understand the “theorem of the cube,” following Mumford’s “Abelian varieties.”

Theorem 1 Let ${X, Y}$ be proper varieties over an algebraically closed field ${k}$, and let ${Z}$ be a connected variety. Let ${\mathcal{L} \in \mathrm{Pic}(X \times Y \times Z)}$ be a line bundle. Suppose there exist ${k}$-valued points ${x_0, y_0, z_0}$ in ${X, Y, Z}$ such that ${\mathcal{L}}$ is trivial when restricted to ${\left\{x_0\right\} \times Y \times Z, X \times \left\{y_0\right\} \times Z, X \times Y \times \left\{z_0\right\}}$. Then ${\mathcal{L}}$ is trivial.

This theorem is extremely useful in analyzing the behavior of line bundles on abelian varieties; see these posts for instance.

I’ve found the proof due to Weil and Murre in the second chapter of Mumford to be quite impenetrable; the argument makes sense line by line but I have never been able to see a larger picture. Fortunately, it turns out that the third chapter has a more scheme-theoretic (nilpotents will be used!) argument which is much more transparent.  (more…)

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.

Last time, we proved an important theorem. Namely, for a proper morphism of noetherian schemes $X \to \mathrm{Spec} A$, we showed that the cohomology of a coherent sheaf $\mathcal{F}$ on $X$, flat over the base, could be described as the cohomology of a finite complex $K$ of flat, finitely generated modules, and moreover that if we base-changed to some other scheme $\mathrm{Spec B}$, we just had to compute the cohomology of $K \otimes_A B$ to get the cohomology of the base extension of the initial sheaf.

With this, it is not too hard to believe that as we vary over the fibers $X_y$, for $y$ in the base, the cohomologies will have somewhat comparable dimensions. Or, at least, their dimensions will vary somewhat reasonably. The precise statement is provided by the semicontinuity theorem.

Theorem 6 (The Semicontinuity theorem) Let ${f: X \rightarrow Y}$ be a proper morphism of noetherian schemes. Let ${\mathcal{F}}$ be a coherent sheaf on ${X}$, flat over ${Y}$. Then the function ${y \rightarrow \mathrm{dim} H^p(X_y, \mathcal{F}_y)}$ is upper semi-continuous on ${Y}$. Further, the function ${y \rightarrow \sum_p (-1)^p H^p(X_y, \mathcal{F}_y)}$ is locally constant on ${Y}$. (more…)