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?
Best Answer
Though I don't have a full answer to your question, the following remarks may help.
Let's distinguish between (1) explicit examples of systems of Diophantine equations that are known to be undecidable, and (2) a system of Diophantine equations that has the property of being undecidable, whether we know it or not.
Regarding number (1), Jones has written down some explicit examples. The main practical difficulty with computing any invariants of these examples is their size. The basic example in the paper has 28 variables and degree $5^{60}$, which can be rewritten as a system in 58 variables and degree 2. My guess is that any Groebner basis algorithm would crash on this example, not because of any particular pathology, but because of its size. However, maybe I'm wrong. The example is written out explicitly so you could try playing with it yourself.
Regarding number (2), although I'm not aware of any precise theorems to this effect, I think that the intuition of most logicians and computability theorists is that "most" systems of Diophantine equations are undecidable. (This is related to the intuition that "most" computably enumerable sets are not computable.) If this intuition is correct, then undecidable Diophantine equations aren't "special"; it's just the reverse—the decidable ones are the ones that are special. Then your question is not very different from the question, what is the geometry of a "random" set of Diophantine equations? So it becomes more of a question in computational commutative algebra than a question in computability theory.