Let $X$ be a complex, projective algebraic variety and assume that $X$ has a model $X_0$ over $\mathbb Z$ i.e. $X\cong X_0\times_{\operatorname{Spec }\mathbb Z}\operatorname{Spec }\mathbb C$. 

Let's take a look at the following quote from the book "Mumford, Oda - Algebraic Geometry II" (In what follows we can put $\mathbb Z=R$): [Check this picture][1]. 

So, the aim of the "Kroneckerian geometry" is to find relations between the (classical) geometry of the complex variety $X$ and the arithmetic (for example Diophantine properties) of the $\mathbb Z$-model $X_0$.

I'd like to collect a partial list of the most important results in this direction. In more practical terms I'm asking you to write the best (or your favourite) examples of "Kroneckerian geometry in action".

Many thanks in advance

  [1]: https://i.sstatic.net/eF0vj.png