Large unions of generalized integral sections on elliptic surfaces

Let $f \colon X \to B$ be a nonisotrivial complex elliptic surface and let $\mathcal{D} \subset X$ be an integral divisor dominating $B$. We study finiteness related properties of generalized $(S, \mathcal{D})$-integral sections $\sigma \colon B \to X$ of $X$. These integral sections $\sigma$ correspond to rational points in $A(K)$ which satisfy the set-theoretic condition $f ( \sigma(B) \cap \mathcal{D})\subset S$, where $S \subset B$ is an arbitrary given subset. For $S \subset B$ finite, the set of $(S, \mathcal{D})$-integral sections of $X$ is finite by the well-known Siegel theorem. In this article, we establish a general quantitative finiteness result of several large unions of $(S, \mathcal{D})$-integral sections in which both the subset $S$ and the divisor $\mathcal{D}$ are allowed to vary in families where notably $S$ is not necessarily finite nor countable. Some applications to generalized unit equations over function fields are also given.