Algebraic Hyperbolicity and Lang Conjecture

Lang猜想说，各种一般类型上的有理曲线和椭圆曲线在代数上是 堕落。我们研究射影空间中一般超曲面的这一猜想。这个问题是 与复双曲性和小林猜想密切相关。在代数设置中，这相当于 在Demailly意义下的代数双曲性。H·克莱门斯证明了一般超曲面是 当超曲面的次数足够大时，代数双曲。我们想要推动他的 关于超曲面到猜想最优的次数的界。我们建议对此进行探讨 通过构造超曲面族的整体喷射空间和研究稳定性来猜想 相应的喷流束的轨迹。

Lang conjecture says that rational and elliptic curves on a variety of general type are algebraically degenerate. We study this conjecture for generic hypersurfaces in projective spaces. This problem is closely related to complex hyperbolicity and Kobayashi conjecture. In the algebraic setting, this amounts to algebraic hyperbolicity in the sense of Demailly. H. Clemens proved that a generic hypersurface is algebraically hyperbolic when the degree of the hypersurface is sufficiently large. We want to push his bound on the degree of hypersurfaces to the conjectured optimum. We propose to approach this conjecture by constructing the global jet spaces of versal families of hypersurfaces and studying the stable loci of the corresponding jet bundles.