As an arithmetic algebraic geometer of the highest moral fiber, I am trained to look at Diophantine equations in terms of the geometry of the corresponding scheme. For instance, if the Diophantine equation comes from a curve, I know that I should compute the genus, and do various things depending on if the genus is $0$, $1$, or $\geq 2$.
But I know very little about what the limits of this geometric approach are. I only know that there exist undecidable Diophantine equations, or families of Diophantine equations. I do not know what their geometry is like!
Do undecidable Diophantine equations, or families of equations, have interesting geometric properties? Can we compute basic geometric invariants like the Hodge diamond, Kodaira dimension, etc.? Are they pathological in every way, or do some of them have properties that might give a naive geometer hope about finding solutions? What would Noam Elkies try to do if he were asked to solve them and did not know they were undecidable, and why would he be stymied?