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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?

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    $\begingroup$ This is bad form -- I should really double-check this first. However, my vague recollection is that inside $\overline{\mathcal{M}}_{g,0}$, the following divisor class, $$ D = \sum_{h=1}^{[g/2]} h(g-h) \Delta_h,$$ is zero on the $F$-curves of the form $\overline{\mathcal{M}}_{0,4}$. Clearly it is not zero on some of the $F$-curves of the form $\overline{\mathcal{M}}_{1,1}$. I will try to double-check this. If it is correct, then I will post it as an answer. $\endgroup$ Jan 7 '14 at 20:45
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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 $\overline{\mathcal{M}}_{1,1}$.

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