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 $\mathcal{O}(D)\cong \mathcal{O}(D')$?



F-curves and divisors on $\overline{\mathcal{M}}_{g,n}$
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](https://mathoverflow.net/questions/98218/)) 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](https://mathoverflow.net/questions/139013/)).


