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In the paper "A non-linear deformation of the Hitchin dinamycal system", Donagi-Ein-Lazarsfeld describe the irreducible components of the moduli space $\mathcal M_R$ of stable sheaves of numerical rank $1$ and degree $k$ on a ribbon $R=2C$ inside a K3 surface. They are $\mathbb{N}_{\infty}$, the moduli space of stable rank $2$ vector bundles on $C$ of degree $k+2-2g$, and $\mathbb{N}_{d}$ for $0 \leq d < 2g-2$ and $d=k$ mod $2$, the stable $\mathcal O_R$-modules $\mathcal E$ whose restriction to $C$ has rank $1$ but it is not locally free in $d$ points of $C$. (See page 10 of the paper for other details).

If I consider the moduli space of semistable sheaves $\overline{\mathcal M}_R$, how can one describe the intersections of its irreducible components?

Any reference on the topic is welcome!

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