# Relationship between virtual cohomological dimension and tautological rings for moduli spaces of curves

Here's the short version of the question. For $M_{g,n}$, $M_{g,n}^{rt}$, $M_{g,n}^{ct}$ and $\overline M_{g,n}$ it seems that the virtual cohomological dimension is given by the complex dimension plus the top degree of the tautological ring. Why??

Here's the long version. Let $2g-2+n > 0$. Let $R^\bullet(\overline M_{g,n})$ be the tautological ring of the moduli space of stable $n$-pointed genus $g$ curves, i.e. the subring of the Chow ring generated by all $\kappa$-classes, $\psi$-classes and classes of boundary strata. This definition looks rather ad hoc but there are more natural ones. If $U \subset \overline M_{g,n}$ is Zariski open we define the tautological ring of $U$ by restriction. We will consider in this question the open subsets $M_{g,n}$ (parametrizing smooth curves), $M_{g,n}^{rt}$ (parametrizing curves with rational tails, i.e. one component has geometric genus $g$ and all other are rational) and $M_{g,n}^{ct}$ (parametrizing curves of compact type, i.e. every node of the curve is separating).

For all $U$ as above one knows the largest $k$ for which $R^k(U)$ is nonzero (the top degree of the tautological ring): this is

• $k=3g-3+n$ for $U = \overline M_{g,n}$
• $k=2g-3+n$ for $U = M_{g,n}^{ct}$
• $k=g-2+n - \delta_{0,g}$ for $U= M_{g,n}^{rt}$
• $k=g-1 + \delta_{0,g} - \delta_{0,n}$ for $U=M_{g,n}$.

This is due to Looijenga, Graber-Vakil, Faber-Pandharipande. Here $\delta_{i,j}$ is the Kronecker delta. In the first three cases (which are perhaps the natural ones to consider) the top degree is in fact one-dimensional.

One can also consider the virtual cohomological dimension (vcd) of all spaces above. The vcd of $M_{g,n}$ was determined by Harer; it is equal to $$4g-4+n + \delta_{0,g} - \delta_{0,n}.$$ The vcd of $\overline M_{g,n}$ is obviously equal to its (real) dimension, $6g-6+2n$. For $M_{g,n}^{ct}$ and $M_{g,n}^{rt}$ the vcd is not known, but there is an upper bound due to Gabriele Mondello: the vcd is bounded by the virtual homotopical dimension (vhd), for which he finds the following bounds: $$\mathrm{vhd}(M_{g,n}^{ct}) \leq 5g-6+2n,$$ $$\mathrm{vhd}(M_{g,n}^{rt}) \leq 4g-5+2n - \delta_{0,g}.$$ Let's assume that these upper bounds for the virtual cohomological dimension are sharp. Then in all four cases above the following MYSTERY EQUATION is true: $$\text{(vcd)} - \text{(top degree of taut. ring)} = 3g-3+n = \text{dimension over \mathbf C}.$$ Is there any plausible reason why something like this should be true?!

• this question has been here since 2013 and i just noticed it. Dan, do you happen to have got an answer since then? – issoroloap Jan 29 '18 at 0:23
• Hi Paolo, maybe. It seems plausible that the relationship between vcd and tautological classes passes through the conjectures of Looijenga on the # of affine open subsets needed to cover the moduli spaces of curves. Namely: (1) knowing what is the top degree of the tautological ring gives an upper bound on the dimension of a compact subvariety of the moduli space, (2) knowing the minimum number of open affines needed in a cover gives an upper bound for both the vcd and the largest dimension of a compact subvariety... – Dan Petersen Jan 29 '18 at 6:30
• ... and if all those bounds are sharp, then the "mystery equation" follows as a consequence. – Dan Petersen Jan 29 '18 at 6:30