A unified finiteness theorem for curves

We study the arithmetic of Galois-invariant sets of points on algebraic curves with controlled reduction behavior. Let $C$ be a smooth projective curve with a smooth proper model $\mathcal{C}$ over ${\mathcal{O}}_{K,S}$. We define ${\mathrm{\Omega }}_n$ as the set of $n$-element subsets of $C(\bar{K})$ that are invariant under $\text{Gal}(\bar{K}\mathrm{/}K)$ and such that no two points in the set become identified modulo any prime $\mathfrak{p}\mathrm{\notin }S$. Our main result establishes that ${\mathrm{\Omega }}_n$ breaks into finitely many orbits under the action of ${\text{Aut}}_{{\mathcal{O}}_{K,S}}(\mathcal{C})$, generalizing finiteness theorems of Birch–Merriman, Siegel, and Faltings.