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 (KX+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,∑i(1−1/mi)Di). 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.
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BibTeX
@misc{chen2026rational,
title = {{On rational curves relative to a divisor}},
author = {Jiun-Cheng Chen and Hsian-Hua Tseng},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05809},
eprint = {2611.05809},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05809},
}
RIS
TY - UNPB
TI - On rational curves relative to a divisor
AU - Jiun-Cheng Chen
AU - Hsian-Hua Tseng
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05809
UR - https://arxiv.org/abs/2611.05809
AB - 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,∑_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.
ER -