The Serre intersection formula, as an alternating sum of contributions from Tor-groups, is something that combines a lot of ingredients that I'm interested in, but I've never really felt that I have a "grip" on it. One of the reasons for this is that, despite making attempts on a couple of occasions, I never seem to have been able to concoct an example where I get contributions from higher Tor-terms that's geometrically satisfying.

This is, unfortunately, a vague description of what I'm looking for. When you describe intersection multiplicity, it's relatively easy to give examples of double and triple intersection of planar curves, give a proto-definition in terms of degree of tangency, show how this is captured by their scheme-theoretic intersection, and talk about what happens when the curves are moved slightly. This is the kind of thing I miss having.

I should also clarify that I'm looking for something slightly more complicated than intersecting a planar curve with a point.

Are there examples of the Serre intersection formula in this vein?