Let $X$ be a K3 surface. Let $E$ be a semistable rank 3 vector bundle. Now suppose $0 = E\_0\subset E\_1\cdots\subset E\_s=E$ be the Harder-Narasimhan filtration. Suppose $E\_1$ is $\mu$-stable and rank $E\_1$ is 2. Then the paper says that $E$ sits in the following short exact sequence :


$0\longrightarrow E\_1\longrightarrow E\longrightarrow N\otimes I\_\xi\longrightarrow 0$ where $N$ is a line bundle and $I\_\xi$ is the ideal sheaf of a 0-dimensional subscheme $\xi\subset X$.


How do we get this exact sequence? What are $N$ and $\xi$ exactly? Any help will be appreciated! Thanks in advance!



Why does this vector bundle on the surface sit in this exact sequence?
Any torsion free sheaf on a smooth surface can be written in the form $N \otimes I\_\xi$, in particular this holds for $E/E\_1$. Explicitly, one can write $N = \det(E)\otimes \det(E\_1)^{-1}$, and also one can figure out the length of $\xi$ from Chern classes of $E\_1$ and $E$. The actual subscheme $\xi$ is more difficult to understand.


