Does a (connected?) projective (or just complete?) variety over a finite field have cardinality congruent to 1 mod the size of the field?
Do (connected?) affine varieties have cardinality a power of the size of the field?
I think the answers to these questions are supposed to be yes, but I suspect some sort of better formulation of the question (missing hypotheses perhaps) is needed. Maybe there needs to be some mention of rational points over an algebraic closure. Feel free to fill such things in.
It would be nice if there was any sort of converse, like "if a variety is connected and has cardinality 1 mod |K|, then it is complete", but I'm less sure something like this exists.