Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?!

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.