TITLE: What is the geometry of an undecidable diophantine equation?
QUESTION [54 upvotes]: 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?

REPLY [16 votes]: This is just a long comment, rather than an answer to Will's question. There are "standard" algorithmically unsolvable problems in combinatorial group theory: Word problem, conjugacy problem, triviality problem for the group, isomorphism problem for a pair of groups, etc. The most fruitful line of arguments relating logic to geometry in group theory is to find geometric conditions on finitely-presented group(s) which are sufficient for solvability of these problems. It turned out that, appropriately defined hyperbolicity is a geometric condition implying algorithmic solvability of all the "standard" problems. Furthermore, it turned out that hyperbolicity is, probabilistically speaking, "generic". Moreover, one of the oldest algorithms for solving word problem (Dehn's algorithm/linear isoperimetric inequality) is one of many equivalent definitions of hyperbolicity. In view of this, I would be looking for algebro-geometric features which are sufficient for algorithmic solvability of Diophantine equations, rather than the other way around. However, since I am neither a number-theorist nor a logician, I do not know what they could be.