I am currently trying to learn a bit about Grothendieck-Riemann-Roch...


To try to get a better feeling for it, I am looking for examples of nice applications of GRR applied to a proper morphism $X \to Y$ where $Y$ is *not* a point. I already I know of a fair number of nice applications of HRR, i.e. GRR when $Y$ is a point. I've read through some of the relevant sections of Fulton's Intersection Theory book, but I've only found applications of HRR there, though it's very possible that I overlooked something.


I am also interested in seeing worked-out, explicit, concrete examples, with explicit Chow/cohomology classes.


Thanks much!



Applications of Grothendieck-Riemann-Roch?
Check out Harris & Morrison's "Moduli of Curves", section 3E. There is a wealth of examples of applications of GRR coming from moduli theory, in which one applies it to projection from the universal family or some fibered power of the universal family. The basic idea in these cases is that both the base space and the total space are rather complicated beasts but the fibers of the morphisms are usually quite tractable, since they are just the gadgets you are trying to parametrize. 


For more examples in the same vein, you could read the classic "Towards an enumerative geometry of the moduli space of curves" by David Mumford.


