While discussing Ticket 16953, which improves functionality for enumerating 
points in toric varieties over finite fields, I said:

"The actual functionality works fine, albeit slowly. I would find it 
helpful if there were a way to save partial progress when computing a point 
set: I'm interested in examples that are large enough that the Sage cloud 
may kill the process before I'm done."

Volker responded:

"I hear you, and I agree that checkpointing is an important feature. But 
its not really clear how that should be implemented or how the interface 
would look like. I guess the iterator is (or can be made) pickleable, but 
really checkpointing needs to be added on the @parallel level. In the short 
run its probably easier to email William to request a longer timeout on 
your project..."

Since checkpointing ideas are beyond the scope of Ticket 16953, I'm moving 
this discussion here.  Some thoughts:

* I'm particularly interested in examples involving hypersurfaces in smooth 
toric varieties over finite fields.  For smooth toric varieties, one can 
stratify the points according to which torus they lie in, which in turn is 
tracked combinatorially by elements of the fan.  Could one report which 
toruses have been checked?  Or make it possible to check only in a certain 
torus?  Presumably most of the time will be spent checking points in the 
big torus, but on the other hand I can imagine somebody might only *care* 
about points in the big torus, so this might be useful functionality to 
separate.

* More naively, should there be an option to report *which* points lie on a 
hypersurface?  For toy examples, seeing some points might be enough to 
guess the rest.  More generally, I can imagine applications where somebody 
might be happy to find *any* points on a hypersurface over a finite field, 
and not care about whether all of them have been identified.  

--Ursula.

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