Extension of a holomorphic vector bundle on a nodal curve

Question

I am reading a paper on holomorphic curves and stuck in an argument about extension of a given holomorphic vector bundle over a nodal curve.

Let $C$ be a nodal curve without closed componets and $E$ a holomorphic vector bundle on $C$. For a compact nodal curve $\tilde{C}$ containing $C$, how can $E$ extend to a holomorphic vector bundle $\tilde{E}$ on $\tilde{C}$? Moreover, in the same paper, the author claims that one can choose $\tilde{E}$ in such a way that $\langle c_{1}(\tilde{E}), \tilde{C}_{i} \rangle$ is sufficiently large for any component $\tilde{C}_{i} \subset \tilde{C}$. Could you please tell me how to take such an extension?

Any hint and comment are really appreciated. Thank you in advance.

Answer 1

First, it is a general fact that a coherent sheaf can be extended from an open subscheme. Indeed, if $j \colon U \to X$ is an open embedding and $F$ is a coherent sheaf on $U$, the quasicoherent sheaf $j_*F$ on $X$ is the union of its coherent subsheaves, so one can write $j_*F = \cup G_\alpha$. Then $F = j^*j_*F = \cup j^*G_\alpha$. Using the noetherian property one obtains $F$ as a finite union of $j^*G_\alpha$, then the corresponding finite union of $G_\alpha$ provides a coherent extension of $F$.

Now, let $G$ be a coherent extension of $E$ to $\tilde{C}$. The torsion subscheaf of $G$ is supported on $\tilde{C} \setminus C$. Therefore the torsion free quotient of $G$ is isomorphic to $E$ on $C$. Assuming for simplicity that $\tilde{C} \setminus C$ consists of smooth points of $\tilde{C}$ we conclude that $\tilde{E}$ is locally free at $\tilde{C} \setminus C$ (because it is torsion free) and also at $C$ (because it coincides with $E$ on $C$). This answers the first question.

Now let $C_0 \subset \tilde{C}$ be one of the components and let $P_0 \in C_0 \setminus C$. Then the bundle $\tilde{E} \otimes \mathcal{O}_{\tilde{C}}(kP_0)$ provides yet another extension of $E$ and its $c_1$ equals $c_1(\tilde{E}) + r\cdot k$, where $r$ is the rank of $E$. So, for $k \gg 0$ it is sufficiently large. So, it remains to apply this construction for every component of $\tilde{C}$.