Italian-style algebraic geometry in homotopy theory?

Question

In homotopy theory, stacks can be occasionally useful (i.e. in the chromatic story). I come from a differential geometry background, so when people say that algebraic geometry is useful in homotopy theory, I have mixed feelings (like the stack classifying formal groups of height $n$ is basically a quotient of a single point, not a particularly geometric object from some perspectives).

Has Italian-style algebraic geometry ever shed light on homotopy theory? A caricaturistic (and probably wrong) example of how an an answer should look like: "27 lines on a cubic surface are actually in bijection with this stable homotopy group of spheres!"

P.S. "Italian-style" can be understood in many ways; one possible interpretation is "the study of non-trivial facts about separated schemes of finite type over $\mathbb{C}$."

Answer 1

This is just a long comment. Homotopy theory is a rather broad field, so the answer to your question depends on what part of homotopy theory you'd like to see having interactions with "Italian-style" algebraic geometry.

I assume that there are more interactions of the sort you desired in motivic homotopy theory (particularly given your definition of "Italian-style" algebraic geometry), e.g., results relating $\mathbf{A}^1$-homotopy theory to birational geometry. I know very little about this area, though, so hopefully somebody with more expertise could elaborate on this. If I recall correctly, Wickelgren gave some lectures about this topic at the Arizona Winter School this year, but their website is down at the moment, so I can't link to the notes.

In chromatic homotopy theory (which you brought up with the example of the moduli of formal groups), however, the situation is different. (Denis brought up the example of elliptic curves, but you said that didn't count.) The reasoning behind this is partly due to the fact that I haven't seen such interactions in the literature, and partly philosophical: most interesting algebro-geometric phenomena in chromatic homotopy theory stem from stacky phenomena. For instance: the $E_2$-page of the homotopy fixed points spectral sequence calculating $\pi_\ast KO$ is exactly the cohomology of the classifying stack $B\mathbf{Z}/2$; the $E_2$-page of the Adams spectral sequence is the cohomology of the classifying stack $B\mathrm{Aut}(\widehat{\mathbf{G}}_a)$. Already at the level of $E_2$-pages (the "purely algebraic" input into homotopy theory), one might say that interesting homotopy theoretic phenomena wouldn't be visible without the introduction of stacks (over $\mathbf{Z}$). (This is obviously an overblown statement, because topologists did calculations way before stacks were introduced into homotopy theory.)