I'm attempting to understand the Bombieri-Lang Conjecture:



> 
> If $X$ is a smooth projective variety of general type defined over a number field, then the set of rational points of $X$ is not dense.
> 
> 
> 


I don't understand what it means for a variety to be ''of general type''. I know it's when the variety's Kodaira dimension is maximal, but this doesn't mean much to me. Is there an equivalent condition, or more intuitive way to visualise Kodaira dimension?



Understanding what it means to be ''of general type''
To start understanding this, it's probably best to start with some examples.


First, the conjecture says that if a curve has Zariski dense rational points, then it is genus zero or one. This is known (Faltings).


Second, the conjecture, plus the Enriques-Kodaira classification, says that if a surface has Zariski dense rational points, then it is a rational surface, a ruled surface, an abelian surface, a K3 surface, an Enriques surface, an elliptic surface, a hyperelliptic surface, or a blow-up of one of these. A general type surface is simply one that is not any of those.


I don't know how many of these are possible to visualize but each of these has a much more definite structure and set of key properties than the class of all general type surfaces.


You could also try to understand general type surfaces through some positive examples like high-degree hypersurfaces, covers of products of two higher genus curves, and hyperbolic surfaces.


