发表机构
Academy of Mathematics and Systems Science, Chinese Academy of Sciences; University of Chinese Academy of Sciences（中国科学院数学与系统科学研究院; 中国科学院大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
通过Poincaré问题与2-jet微分叶状结构，证明$\mathbb{P}^3$中次数至少18的一般曲面无有理或椭圆曲线，进而建立其Kobayashi双曲性，并推广到平面曲线补集。
AI 中文摘要
我们证明了$\u001b[3m\u001b[0m\mathbb{P}^3$中次数至少为$18$的一般曲面不包含有理曲线或椭圆曲线，这加强了Clemens的经典结果，将原来的“非常一般”假设替换为真正的Zariski开条件。此前，在“一般”情形下的不存在性结果仅在更高次数时已知。结合已有的整曲线代数退化结果，我们推导出$\mathbb{P}^3$中次数至少为$18$的一般曲面的Kobayashi双曲性，从而解决了Demailly–El Goul提出的一个问题。我们的证明使用由$2$-jet微分诱导的叶状结构。两个独立的此类微分产生一个多重叶状结构，与所有有理曲线和椭圆曲线相切。我们为其代数叶建立了Poincaré型界，从而提供了一种将非常一般性结论提升为一般性结论的机制。我们的方法也适用于平面曲线的补集。特别地，我们证明了$\mathbb{P}^2$中两条一般三次曲线的补集是双曲嵌入的。
英文摘要
We prove that a general surface in $\mathbb{P}^3$ of degree at least $18$ contains no rational or elliptic curves, strengthening the classical result of Clemens by replacing the original very general assumption by a genuine Zariski-open condition. Previously, nonexistence results in the ``general'' setting were known only in much higher degrees. Combining this with established algebraic degeneracy results for entire curves, we deduce the Kobayashi hyperbolicity of a general surface in $\mathbb{P}^3$ of degree at least $18$, thereby resolving a question asked by Demailly--El Goul. Our proof uses foliations induced by $2$-jet differentials. Two independent such differentials give rise to a multi-foliation tangent to all rational and elliptic curves. We establish a Poincaré-type bound for its algebraic leaves, yielding a mechanism to upgrade very general statements to general ones. Our method also applies to complements of plane curves. In particular, we prove that the complement of two general cubic curves in $\mathbb{P}^2$ is hyperbolically embedded.
Comments36 pages
