abstract:

We prove a generalized version of Schmidt's subspace theorem for closed subschemes in general position in terms of suitably defined Seshadri constants with respect to a fixed ample divisor. Our proof builds on previous work of Evertse and Ferretti, Corvaja and Zannier, and others, and uses standard techniques from algebraic geometry such as notions of positivity, blowing-ups and direct image sheaves. As an application, we recover a higher-dimensional Diophantine approximation theorem of K.~F.~Roth-type due to D.~McKinnon and M.~Roth with a significantly shortened proof, while simultaneously extending the scope of the use of Seshadri constants in this context in a natural way.

摘要：

我们针对固定位置的充分除数，通过适当定义的Seshadri常数，证明了一般位置上封闭子方案的Schmidt子空间定理的广义形式。我们的证明建立在Evertse和Ferretti，Corvaja和Zannier等人的先前工作之上，并使用了代数几何的标准技术，例如正性，爆炸和直接图像滑轮等概念。作为应用，我们恢复了D.〜McKinnon和M.〜Roth导致的K.〜F.〜Roth型高维Diophantine逼近定理，同时大大缩短了证明，同时扩展了Seshadri的使用范围在这种情况下，自然地使用常量。