I am not good at algebraic geometry and almost surely am misunderstanding something.
Got an alleged argument against Bombieri-Lang conjecture and would like to know what the mistake is.
One of the most simplest formulations of Bombieri-Lang conjecture is in Joe Silverman's answer on MO, paraphrasing
...For a surface of general type ... the Bombieri-Lang conjecture says that the solutions in rational numbers ...are.. not Zariski dense (lie on a finite set of curves).
Take the affine surface over the rationals:
$$ z^6 + x^4 - y^2=0$$
According to Magma it is of general type.
Fix $z$ squarefree integer $k$. This gives quartic model of elliptic curve $k^6+x^4-y^2=0$.
This is birationally equivalent to Weierstrass $$v^2=-4k^6u+u^3 $$
As $k$ varies, this gives infinitely many positive rank elliptic curves.
What is wrong with this alleged contradiction?
Comment suggests it might be not of general type.
In Magma online: http://magma.maths.usyd.edu.au/calc/
K<x,y,z,t>:=ProjectiveSpace(Rationals(),3);
p:=z^6 + x^4*t^2 - y^2*t^4;
S:=Surface(K,p);
KodairaEnriquesType(S);
KodairaDimension(S);
//returns:
// 2 0 General type
// 2
Added The main misunderstanding was caused by incorrect usage of
Magma function. The correct way to check is KodairaEnriquesType(S : CheckADE := true); because of certain assumptions, but this might take much longer.
With this change I get error and no result.