I tried to answer an earlier question as to uses of GRR, just from my reading, although i do not understand GRR. Today i tried to understand the possible idea behind GRR. After editing my answer accordingly, it occurred to me i was asking a question instead of giving an answer. My question is roughly whether the following speculation is in the ball park as to the purpose of GRR.

I've been thinking about Riemann Roch today, and reading Riemann. After dealing with a fixed divisor D, Riemann observes that his result proves every divisor of degree g+1 dominates the pole divisor of a non constant meromorphic function. Then he says that it may be possible to find a special divisor of even lower degree that dominates the poles of a non constant function. I.e. he begins to vary the divisor. By a rank calculation he shows one cannot expect a non constant function unless the pole divisor has degree at least (g/2)+1.

Now following his lead, we are led to vary the curve instead of the divisor. E.g. we might consider the family of curves over the moduli space. Then a good Riemann Roch theorem should let us relate the riemann roch theorem for the curve fibers, to a conclusion for a related sheaf on the base space., like a kunneth type formula, relating cohomology of base space total space and fiber.

I.e. a nice divisor like the canonical divisor on a curve, should be cut out on each curve fiber by a divisor on the total space, by intersecting it with each curve. (e.g. we could restrict the sheaf O(1) on the plane, to every curve of degree 4.) then we can push this sheaf from the total space down to the base space, i.e. the moduli space of curves. A good relative riemann roch theorem would then relate the universal canonical sheaf on the total space, to the canonical sheaves on the curve fibers, and the cohomology of the push down of the universal sheaf to the base space, the moduli space of curves.

Ideally such a relation would let one compute invariants of sheaves on the moduli space that do arise by pushing down sheaves on the total space of curves. Hopeful applications might include finding ample sheaves on Mg, hence proving projectivity, and computing invariants of the canonical sheaf on Mg, hence potentially estimating the kodaira dimension.

Now this is all speculation since i do not understand even the statement of the GRR, and have not read the paper of Harris-Mumford in which the application i cited above is made. Moreover I have never seen any proof of kodaira dimension of Mg using this method. Perhaps someone more knowledgable will comment on these speculative applications?

Is this roughly the idea behind GRR and Mumford's applications of it? I.e. is the idea of GRR to understand the cohomology of a sheaf on a base space which arises as a push down, by restricting it to the fibers of the map? and how helpful is this in practice?

specific question: if chi(O) is constant on fibers, does GRR allow one to determine chi(O) of either total space or base space from the other?