F-curves and divisors on $\overline{\mathcal{M}}_{g,n}$

Question

It is conjectured that a divisor on $\overline{\mathcal{M}}_{g,n}$ would be ample if $D\cdot C >0$ for all F-curves $C\subset \overline{\mathcal{M}}_{g,n}$. Does the intersection degree with all F-curves identify completely a divisor $D$? I.e.: if $D$ and $D'$ have the same degree once restricted to any F-curve, do we have $\mathcal{O}(D)\cong \mathcal{O}(D')$?

Answer 1

Yes, this is true.

First of all, the map $\newcommand{\MM}{\mathcal{\overline M}_{g,n}}A^1(\MM) \to H^2(\MM)$ is an isomorphism (say with $\mathbf Q$-coefficients on both sides), so by Poincaré duality it will be enough to prove that $H^{6g-6+2n-2}(\MM)$ is spanned by classes of F-curves.

We do this using the spectral sequence associated to a filtration. Filter $\MM$ by taking $X_k$ to be the union of all strata of dimension $\leq k$ in the stratification by topological type, so we have a chain of closed subvarieties $$ \emptyset = X_{-1} \subset X_0 \subset \cdots \subset X_{3g-3+n} = \MM.$$ I always find it easier to think about the Poincaré dual spectral sequence in compactly supported cohomology, which reads $$ E_1^{pq} = H^{p+q}_c(X_p \setminus X_{p-1} ) \implies H^{p+q}_c(\MM).$$ Each $X_p \setminus X_{p-1}$ is a disjoint union of strata, each of the form $\prod \mathcal M_{g',n'}$ (possibly divided by a finite group). We want to show that $H^2_c(\MM)$ is spanned by F-curves. Now $H^2_c(X_0)$ obviously vanishes. $H^2_c(X_1 \setminus X_0)$ is given by the fundamental classes of all 1-dimensional strata. These are by definition the F-curves. For the 2-dimensional strata we have $H^2_c(\mathcal M_{0,5}) \neq 0$ and $H^2_c(\mathcal M_{0,4}\times \mathcal M_{0,4}) \neq 0$, but their Hodge structures are not pure (see here) so these classes can't survive to $E_\infty$ (because the cohomology of $\MM$ has a pure Hodge structure). It's easy to check that $H^2_c$ vanishes for all other 2-dimensional strata. Finally I claim that also $H^2_c(X_k \setminus X_{k-1})$ vanishes for all $k > 2$. This is (for instance) an easy consequence of the virtual cohomological dimension of $\mathcal M_{g,n}$ (see here).