2 problems

Let $X$ be a compact Riemann surface of genus $g>1$ with its Arakelov metric, and let $\gamma$ be a simple separating curve dividing $X$ into subsurfaces $X_{+}$ and $X_{-}$. In th…

Let $\pi: X\rightarrow B=\operatorname{Spec}(\mathcal{O}_{K})$ be an integral, flat, projective scheme of dimension $2$, and let $L$ be a metrized line bundle on $X$. Suppose there…