Lifting divisors with imposed ramifications on a generic chain of loops

Abstract:

Let $C$ be a curve over an algebraically closed non-archimedean field with non-trivial valuation. Suppose $C$ has totally split reduction and the skeleton $\Gamma$ is a chain of loops with generic edge lengths. Let $P$ be the rightmost vertex of $\Gamma$ and let $\mathcal {P}\in C$ be a point that specializes to $P$. We prove that any divisor class on $\Gamma$ with imposed ramification at $P$ that is rational over the value group of the base field lifts to a divisor class on $C$ that satisfies the same ramification at $\mathcal {P}$, which extends the result in [Canad. Math. Bull. 58 (2015), 250–262].