Still at UGA -- I just saw a great talk by Jason van Zelm (a student of Nicola Pagani who apparently does not have a webpage), constructing non-tautological cycles on \(\overline{\mathscr{M}_g}\) for \(g\geq 12\). As in Graber-Pandharipande's construction of non-tautological classes, the key point is that the class of the diagonal $$[\Delta] \in H^*(\overline{\mathscr{M}_{g, n}}\times \overline{\mathscr{M}_{g, n}}, \mathbb{Z}) \not\in R_{g, n}^*\otimes R_{g,n}^*$$ for large \(g, n\), where \(R_{g,n}\) denotes the tautological ring of \(\overline{\mathscr{M}_{g,n}}\) (in cohomology). As in Graber-Pandharipande, one then shows that the pullback of some natural cycle (constructed via admissible covers -- in this case, curves admitting a \(2:1\) map to an elliptic curve) pulls back to the diagonal under a gluing map, and use that these pullbacks send tautological cycles to elements of \(R_{g,n}^*\otimes R_{g,n}^*\). In this case, we look at the gluing map $$\overline{\mathscr{M}_{g,1}}\times \overline{\mathscr{M}_{g, 1}}\to \overline{\mathscr{M}_{2g}}.$$

This is crazy to me. One of the standard definitions of the tautological ring(s) is as the smallest collection of subrings \(R_{g,n}^*\subset A^*(\overline{\mathscr{M}_{g, n}})\) preserved under pullback and pushforward through a natural collection of maps (gluing, forgetting a point, etc.). It seems natural to me to ask for tautological rings \(R_{g, n, m}\subset A^*(\overline{\mathscr{M}_{g, n}^n})\) which contain diagonal cycles and are stable under pullbacks and pushforwards through natural maps. Jason has suggested to me that this collection might be the same as that obtained by adding cycles obtained through pullback and pushforward along various natural maps involving the moduli of admissible maps. This is also a natural idea, and if he can prove something like this, I think it would be very exciting. I have no idea how to prove it, personally.

One appealing aspect of these ideas is that there is another notion of "tautological ring," which is almost exactly defined to deal with this sort of thing. Namely, Qizheng Yin, in this paper on K3s, defines a natural system of "tautological rings" for any variety. Let \(X\) be a variety and let \(R^*_n\subset A^*(X^n)\) be the smallest system of subrings stable under pullback and pushforward through the natural maps \(X^n\to X^m\) (i.e. projections and generalized diagonal maps -- to be explicit, all the maps obtained in the Cech hypercover associated to the map \(X\to *\)). What Graber-Pandharipande and now Jason's result suggest to me is that we should merge these two notions of tautological ring.

Jason rightly points out that maybe this enlarged tautological ring is incomputable (rather, even less computable than the usual tautological ring). But apparently he can compute with it well enough to prove his very nice result, so who's to say?