Restriction of sheaves on curves

Question

Let $C$ be a scheme of pure dimension $1$. Let $C_1$ be a closed subscheme of $C$ of pure dimension $1$. Denote by $i:C_1 \hookrightarrow C$ a closed immersion. Given a sheaf $\mathcal{F}$ on $C$, there is a natural morphism $\mathcal{F} \to i_*(i^{-1}(\mathcal{F}))$. Suppose that $\mathcal{F}$ is a locally free $\mathcal{O}_C$-module. Is it true that $H^0(\mathcal{F}) \to H^0(i_*(i^{-1}(\mathcal{F})))$ is surjective? If not, is there any additional condition we can impose on $C_1$ (for example $C\backslash C_1$ is affine) such that we get a positive answer to the question?

Answer 1

There are counterexamples. Let $C$ be a union of two copies, $C_1$ and $C_2$, of $\mathbb{P}^1$ intersecting transversally at a single ordinary node, $p$. For definiteness, in $\mathbb{P}^2$ with homogeneous coordinates $[x,y,z]$, let $C$ be the zero scheme of $xy$, let $C_1$ be the zero scheme of $x$, let $C_2$ be the zero scheme of $y$, and let $p$ be $[0,0,1]$. Let $q$ be a point on $C_2\setminus (C_1\cap C_2)$, e.g., $q$ could be $[1,0,0]$. Let $\mathcal{F}$ be the ideal sheaf of $q$ as a closed subscheme of $C$. In particular, on the open subscheme $U=C\setminus\{q\}$, $\mathcal{F}$ is isomorphic to the structure sheaf $\mathcal{O}_U$. Thus the image of $1\in \mathcal{O}_U(U)$ gives a nonzero global section of $i^{-1}(\mathcal{F})$ on $C_1$, hence also of $i_*(i^{-1}(\mathcal{F}))$ on $C$. On the other hand, $\mathcal{F}$ has only the zero global section.

The complement of $C_1$ in $C$ is affine, so that hypothesis is not sufficient to conclude surjectivity.