  Determining all rational points on a curve of genus at least 2 can be
difficult. Chabauty's method (1941) 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. If the Mordell-Weil
rank is less than the genus then this method has never failed.
  Minhyong Kim's non-abelian Chabauty programme aims to remove the condition on
the rank. The simplest case, called quadratic Chabauty, was developed by
Balakrishnan, Dogra, Mueller, 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).
