The Picard rank of a K3 surface counts the number of independent algebraic curves on it — the rank of its Néron–Severi group. In a family of K3 surfaces (a continuously varying collection), the generic fiber has some Picard rank, and special fibers may have higher rank. “Jumping” means the Picard rank increases at special parameter values.
The paper on Picard rank jumps in positive characteristic (arXiv: 2603.22678) proves that for a non-isotrivial family over a curve in characteristic p > 2, infinitely many fibers have Picard rank strictly greater than the generic rank — provided the family avoids a Frobenius obstruction and has big monodromy.
The Frobenius obstruction is characteristic of positive characteristic: the Frobenius endomorphism can force all fibers to have the same Picard rank, preventing any jumping. This is a structural impossibility — no monodromy argument can overcome it. But when this obstruction is absent and the monodromy is big (the Galois representation on cohomology has large image), jumps must occur, and they occur infinitely often.
The through-claim: Picard rank jumping is generic in positive characteristic, blocked only by Frobenius. The family wants to jump — the monodromy creates pressure for special fibers to exist. The only barrier is the Frobenius endomorphism, which can lock all fibers at the same rank. When Frobenius doesn't obstruct, the pressure wins, and infinitely many fibers jump.
2603.22678. Algebraic geometry / K3 surfaces / Picard rank / positive characteristic / monodromy.