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?



Koszul complex of a variety inside a product
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 $X\_m$.


