In general, how does biodiversity vary across the biosphere?

Life needs energy, water, heat and nutrients. Reduce the availability of one or more of those things and biodiversity generally decreases.

Tropical forests tend to be more diverse than polar ice caps. However, arctic seas tend to be more biologically interesting than the middle of the Sahara. I bet this is what your homework is looking for but it's important to remember there are counterexamples.