  The analogy between the arithmetic of varieties over number fields and the
arithmetic of varieties over function fields is a leading theme in arithmetic
geometry. This analogy is very powerful but there are some gaps. In this note
we will show how the presence of isotrivial varieties over function fields (the
analogous of which do not seems to exist over number fields) breaks this
analogy. Some counterexamples to a statement similar to Northcott Theorem are
proposed. In positive characteristic, some explicit counterexamples to
statements similar to Lang and Vojta conjectures are given.

Algebraic Geometry