Inside the moduli space of curves $\overline{\mathcal{M}}\_{g,n}$ one can distinguish two classes of $F$-curves isomorphic to $\mathbb{P}^1$: those of type $\overline{\mathcal{M}}\_{0,4}$, and those of type $\overline{\mathcal{M}}\_{1,1}$. Are there divisor classes that are trivial once restricted to one or the other (or both) types of $F$-curves?



divisors on $\overline{\mathcal{M}}_{g,n}$ that are trivial on certain $F$-curves
The boundary divisor $\Delta\_0$ has intersection number $0$ with every $F$-curve of the form $\overline{\mathcal{M}}\_{0,4}$, yet has nonzero intersection number with every $F$-curve of the form $\overline{\mathcal{M}}\_{1,1}$. 


**Edit.** Also, since $\lambda$ is the pullback of a divisor class from the Satake compactification of $\mathcal{A}\_g$, also $\lambda$ has intersection number $0$ with every $F$-curve of the form $\overline{\mathcal{M}}\_{0,4}$, yet has nonzero intersection number with every $F$-curve of the form $\overline{\mathcal{M}}\_{1,1}$.


