Let $E\to X$ and $F\to Y$ be vector bundles over (sufficiently nice) surfaces or 3-folds.
If I know that a divisor $D \subset X\times Y$ is the zero locus of a pairing $\alpha\colon E \otimes F \to \mathbb{C}$, can I say something about the first Chern class of the divisor $c_1(D)$?