Koszul complex of a variety inside a product

Question

Suppose $X$ is a smooth projective complete intersection contained in the product $\mathbb{P}^n \times \mathbb{P}^m$, and call $X_n$ and $X_m$ the images of $X$ inside $\mathbb{P}^n$ and $\mathbb{P}^m$ via the two natural projections. Suppose moreover that I have the two Koszul complexes of $X_n\subset \mathbb{P}^n$ and $X_m\subset \mathbb{P}^m$. Can I reconstruct the Koszul complex of $X$? Under what hypotheses?

Answer 1

In general, you can't. By taking quotients, knowing the Koszul complex of $X$ tells you $X$. But there are many subvarieties of $X_n \times X_m$ that project surjectively onto $X_n$ and $X_m$. If $X_n$ and $X_m$ are both positive-dimensional, then the intersection of $X_n \times X_m$ with any hypersurface of bidegree $(a,b)$ for $a>0$ and $b>0$ will map surjectively onto $X_n$ and $X_m$, since by intersection theory its intersection with $pt \times X_m$ and $X_n \times pt$ is nontrivial.

One additional assumption that works is to assume that $X_n$ or $X_m$ is a point. But this seems silly! The most reasonable additional assumption I can come up with is that $X = X_n \times X_m$. There are other lists of assumptions that also give a unique $X$, but those just seem contrived. For instance, you could assume that the projections $X \to X_n$ and $X \to X_m$ are isomorphisms, and that $X$ has no automorphisms, but you are unlikely to get a nice description of the Koszul complex.

For $X = X_n \times X_m$, there is a nice description of the complex. Since we get a set of generators for the ideal of $X_n \times X_m$ by taking the union of a set of generators of $X_n$ and a set of generators of $X_m$, you get the Koszul complex by just tensoring the Koszul complexes of $X_n$ and $X_m$.