Let $\mathbb{E}\rightarrow\overline{M}\_{g,n}$ be the Hodge bundle. Let us cosider an $F$-curve of type $\overline{M}\_{1,1}\subseteq\overline{M}\_{g,n}$. **Is the degree of the restriction of $\mathbb{E}$ on such a curve negative?**



Hodge bundle on F-curves
No. It's easy to compute that the restriction of $\mathbb E\_g$ to $\overline M\_{1,1}$ is an extension of the Hodge bundle $\mathbb E\_1$ on $\overline M\_{1,1}$ and $g-1$ copies of the trivial line bundle. The Hodge bundle on $\overline M\_{1,1}$ has degree $1/24$. 


More generally, it is actually true that $\mathbb E$ is nef.


