# What is the kernel of $i^*:H^*(\overline{\mathcal{M}}_{g,n},\mathbb{Q}) \to H^*(\overline{\mathcal{M}}_{g-1,n+2},\mathbb{Q})$?

Let $$\overline{\mathcal{M}}_{g,n}$$ be the Deligne-Mumford moduli space of stable algebraic curves. I would like to know which rational cohomology classes or at least which tautological classes on $$\overline{\mathcal{M}}_{g,n}$$ vanish when restricted to the divisor $$\delta_{\mathrm{irr}}$$ of curves with at least one non-separating node.

An example would be the top Hodge class $$\lambda_g\in H^{2g}(\overline{\mathcal{M}}_{g,n})$$.

I know that Arbarello and Cornalba proved here that $$i^*: H^*(\overline{\mathcal{M}}_{g,n},\mathbb{Q})\to H^*(\overline{\mathcal{M}}_{g-1,n+2},\mathbb{Q})$$ is injective for $$k\leq2g-2$$ for $$g\leq 7$$ and for $$k\leq g+5$$ for $$g\geq 7$$.

I also understand that it's probably irrealistic to expect an explicit and general description of $$\ker i^*$$, either in cohomology or the tautological ring.

So here is a perhaps more approachable question:

Is $$\ker i^* \subset RH^*(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$$ strictly bigger than the ideal generated by $$\lambda_g$$? I guess so! If yes, can I get some examples, maybe something working in any genus? For instance, what is there in cohomological degree between the bounds of Arbarello and Cornalba and $$2g$$?

• Since cohomology is contravariant, did you intend to write $i^*$ in the opposite direction in your title and on Line 5? Also, since the "Satake contraction" of $\overline{\mathcal{M}}_{g,n}$ has positive-dimensional fibers on the image of $i$, you can produce elements in the kernel of the (contravariant) map $i^*$ by pulling back classes of cohomological degree $2(3g-4)$ by the Satake compactification, e.g., an appropriate power of $\lambda_1$. Is there reason to believe those classes are in the ideal generated by $\lambda_g$? – Jason Starr Jan 29 '18 at 3:25
• At one point Hain and Looijenga conjectured that the ideal inside $R^\bullet(\overline M_{g,n})$ consisting of classes restricting trivially to the Deligne-Mumford boundary is principal, generated by $\lambda_g\lambda_{g-1}$. But it's not clear to me if this conjecture should be believed, this was part of a proposed generalization of the Faber conjecture. The analogue of this conjecture for "compact type" would be that the ideal of tautological classes restricting trivially to $\delta_{irr}$ is generated by $\lambda_g$; again, it's not clear if this should be believed. – Dan Petersen Jan 29 '18 at 6:14
• @JasonStarr, oops did I write it in reverse? I guess Dan edited it. Thanks Dan. – issoroloap Jan 29 '18 at 9:14
• @DanPetersen, yes, I have the same feeling. For instance, imagine you want to define a sort of "compact type COhFT" replacing the loop axiom with the fact that the classes vanish on $\delta_{irr}$. Then the natural modification of the Givental group simply sums over stable trees instead of any stable graph. This action commutes with multiplying by "\lambda_g", so you can't get out of that ideal. This is how I got to asking this question. – issoroloap Jan 29 '18 at 9:14