Things are getting serious: I’m scheduling my qual for September, and so there is an extra incentive to dutifully fill in some of the details of the computations in toric intersection theory from my recent post on Hirzebruch surfaces.
I’ve often found that toric geometry is capable of introducing concepts from general algebraic geometry with the additional grounding of its proximity to combinatorics a valuable aid to visualisation that can help shed intuition on the original notion. I’m undecided as to whether or not intersection theory is such a topic. To me at least it is already quite tangible, and the extensive formulae appearing in the toric setting only really obscure what is happening. The existence of such formulae, on the other hand, is extremely powerful, especially since they depend exclusively on readily available combinatorial data.
There a few general principles that these arguments seem to follow. Of primary use is the fact that affine toric varieties have no nontrivial Cartier divisors, and so the open affine cover coming from the maximal cones of a fan provides a trivialisation for any Cartier divisor.
Another slightly stealthier theme is that there’s very little difference between the orbifold (or simplicial) and the smooth cases. Much like computations in weighted projective space where some weight adjustment is the only change needed to extend calculations done in usual projective space, some weight or ‘multiplicity’ adjustment is all that’s needed.