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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.

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Thanks. Paul and David's responses point out the basic motivation I seem to have overlooked, and so they were very helpful. The further details / examples in A.J.'s response, including the paper link, are very much appreciated. –  StS Oct 25 '09 at 15:40
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The reasons for caring which occur to me seem to fall into two broad categories.

First, line bundles are useful for identifying interesting substacks of your moduli stack. You usually first encounter this idea in GIT theory, but the philosophy applies more generally. See, for example, Teleman's papers on the stack of G-bundles on a curve, and Jarod Alper's paper on "good" moduli spaces ( http://math.columbia.edu/~jarod/good_moduli_spaces.pdf ).

Second, sections of line bundles are a good substitute for/generalization of functions. For example, on the moduli stack of curves, the only holomorphic functions are constant, but there are plenty of meromorphic functions, which are naturally thought of as sections of line bundles. This point of view shows up, for example, in the theory of modular forms (where it clarifies their transformations under SL(2,z)) and in the Beilinson-Bernstein approach to representation theory (where you use "twisted D-modules", which act on sections of a line bundle on the flag variety rather than on functions, to get representations with non-trivial central character).

Is that roughly what you were wanting?

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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.

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Or why would you want line bundles on anything? If you wanted to show it was projective, for instance, you would want an ample line bundle. In general, I think a lot of the motivation for all the technical studying of divisors on M_{g,n} is to try to understand its birational geometry.

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