Assume we have a $K3$-surface $X$ over $\mathbb{C}$ and two rational curves $C_1$ and $C_2$ on $X$ with $C_1.C_2=1$ and $C_i^2=-2$.

Let $x$ be a closed point on the reducible curve $C_1\cup C_2$. We have $H^0(X,O_X(C_1+C_2))=1$, so the pair $(x,O_X(C_1+C_2)$ has the Cayley-Bacharach property, i.e. there exists a vector bundle $E_x$ given by the extension, ($I_x$ is the ideal sheaf of $x$):

$0\rightarrow O_X\rightarrow E_x\rightarrow I_x\otimes O_X(C_1+C_2)\rightarrow 0$

That is $E_x$ is a bundle of rank 2 with $c_1(E_x)=C_1+C_2$ and $c_2(E_x)=1$.

Now given $x\neq y \in C_1\cup C_2$ is it possible to decide whether the bundles $E_x$ and $E_y$ are isomorphic or not? Maybe one should look at the cases $x,y\in C_i$ and $x\in C_1 , y\in C_2$ separately? Does the bundle $E_z$ with $z\in C_1\cap C_2$ have special properties?

Can one decide if there are points such that the bundles $E_x$ admit a representation of the form $0\rightarrow O_X(C_i) \rightarrow E_x\rightarrow O_X(C_j)\rightarrow 0$ with $i\neq j$?

More vaguely: how strong does $E_x$ depend on the point $x$? Is it uniquely determined by it?


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