TITLE: Line bundles on moduli spaces
QUESTION [10 upvotes]: This is perhaps too broad or vague (or silly) a question, but here it is anyway: why should I care about constructing line bundles on a moduli space?  This comes up all of the time, but I seem to be missing the (probably obvious) motivation.
Ideally, it would be nice to attach to this question a particular moduli space (vector bundles on a curve, instantons, etc.), but I think I will leave the task of finding an efficient yet instructive example to someone with more knowledge.
Thanks.

REPLY [4 votes]: In general Cohomology is a tool to linearize global properties (just like calculus is a tool to linearize local properties). 
Line bundles are elements in the (probably) most important cohomology group you have on the space - the Picard group. Depending how you construct them you can know a lot of things about the NEF cone of the space and it's Kodaira dimension. 
So, assuming that moduli space are interesting (they are because you don't have so many ways to construct concrete objects), and that the invariants I mentioned are (they are because they are almost the only ones we know for general spaces), it is interesting.