Since Faltings proved Mordell’s conjecture in [16] in 1983, we have known that the sets of rational points on curves of genus at least $2$ are finite. Determining these sets in individual cases is still an unsolved problem. Chabauty’s method (1941) [10] is to intersect, for a prime number p, in the p-adic Lie group of p-adic points of the Jacobian, the closure of the Mordell–Weil group with the p-adic points of the curve. Under the condition that the Mordell–Weil rank is less than the genus, Chabauty’s method, in combination with other methods such as the Mordell–Weil sieve, has been applied successfully to determine all rational points in many cases.

Minhyong Kim’s nonabelian Chabauty programme aims to remove the condition on the rank. The simplest case, called quadratic Chabauty, was developed by Balakrishnan, Besser, Dogra, Müller, Tuitman and Vonk, and applied in a tour de force to the so-called cursed curve (rank and genus both $3$ ).

This article aims to make the quadratic Chabauty method small and geometric again, by describing it in terms of only ‘simple algebraic geometry’ (line bundles over the Jacobian and models over the integers).

自从 Faltings 在 1983 年证明了 [16] 中的 Mordell 猜想后，我们就知道亏格至少为$2$ 的曲线上的有理点集是有限的。在个别情况下确定这些集合仍然是一个未解决的问题。Chabauty 的方法 (1941) [10] 是对于质数p，在雅可比矩阵的p进点的p进李群中，Mordell-Weil 群的闭包与p进点的闭包相交曲线。在 Mordell-Weil 秩小于属的情况下，Chabauty 的方法结合其他方法，如 Mordell-Weil 筛，已成功应用于许多情况下确定所有有理点。

Minhyong Kim 的 nonabelian Chabauty 计划旨在消除排名条件。最简单的情况称为二次 Chabauty，由 Balakrishnan、Besser、Dogra、Müller、Tuitman 和 Vonk 开发，并在绝技中应用于所谓的诅咒曲线（秩和属均为 $3$ ） 。

本文旨在通过仅根据“简单代数几何”（雅可比矩阵上的线束和整数上的模型）对其进行描述，使二次Chabauty 方法再次变得小而几何。