On rational curves relative to a divisor

Abstract

Let $X$ be a smooth complex projective variety and $D$ a reduced simple normal crossings divisor. We prove that, given a complete curve of negative $(K_X+D)$-degree, through its general points there is a rational curve whose normalization meets $D$ in at most one point. The proof uses ordinary weighted jets of a smooth family constructed from $(X,D)$, a positive parameter line bundle, and an intersection calculation on a ruled surface. We obtain an explicit degree bound depending on a projective embedding. We apply this construction to a cyclic cover compactification associated with a smooth Campana orbifold $(X,\sum_i(1-1/m_i)D_i)$. If its anticanonical divisor is big, we show that there is a covering family of free divisible Campana rational curves with fixed numerical class and contact data. In Picard number one case, the orbifold is Campana rationally connected.