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arXiv · 2609.28465

Constructing Rational Curves via Jets on Projective Varieties with Non-Nef Canonical Bundle

Bin Dong·Guoxiong Gao·Bin Guo·Zeming Sun·Bin Wu·Song-Yan Xie
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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 $\mathbb C$. 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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Bin Dong, Guoxiong Gao, Bin Guo, Zeming Sun, Bin Wu, Song-Yan Xie. 2026-09-23. Constructing Rational Curves via Jets on Projective Varieties with Non-Nef Canonical Bundle. https://arxiv.org/abs/2609.28465

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