Constructing Rational Curves via Jets on Projective Varieties with Non-Nef Canonical Bundle
Abstract
We give an algebraic proof in characteristic zero of the Miyaoka--Mori criterion: every point of a curve of negative canonical degree on a smooth projective variety lies on a rational curve. Our jet technique gives, in addition, an effective numerical decomposition of the original curve class. For each prescribed point, the decomposition contains a rational curve through that point, with a positive coefficient independent of the point and with anticanonical degree at most dim X+1. Together with BDPP cone duality, this recovers the projective uniruledness criterion over ℂ. The main result of this paper was obtained using the Pharos system. A detailed report on the use of Pharos and on the Lean~4 formalization of the main arguments is given in the Appendix B, written by Bin Dong, Guoxiong Gao, Zeming Sun, and Bin Wu.
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BibTeX
@misc{dong2026constructing,
title = {{Constructing Rational Curves via Jets on Projective Varieties with Non-Nef Canonical Bundle}},
author = {Bin Dong and Guoxiong Gao and Bin Guo and Zeming Sun and Bin Wu and Song-Yan Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28465},
eprint = {2609.28465},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28465},
}
RIS
TY - UNPB
TI - Constructing Rational Curves via Jets on Projective Varieties with Non-Nef Canonical Bundle
AU - Bin Dong
AU - Guoxiong Gao
AU - Bin Guo
AU - Zeming Sun
AU - Bin Wu
AU - Song-Yan Xie
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28465
UR - https://arxiv.org/abs/2609.28465
AB - We give an algebraic proof in characteristic zero of the Miyaoka--Mori criterion: every point of a curve of negative canonical degree on a smooth projective variety lies on a rational curve. Our jet technique gives, in addition, an effective numerical decomposition of the original curve class. For each prescribed point, the decomposition contains a rational curve through that point, with a positive coefficient independent of the point and with anticanonical degree at most dim X+1. Together with BDPP cone duality, this recovers the projective uniruledness criterion over ℂ. The main result of this paper was obtained using the Pharos system. A detailed report on the use of Pharos and on the Lean~4 formalization of the main arguments is given in the Appendix B, written by Bin Dong, Guoxiong Gao, Zeming Sun, and Bin Wu.
ER -