# Degeneration of a fiber of a rational elliptic surface

Let $$\mathcal{E}$$ be a rational elliptic surface, obtained by blowing up nine base points of a pencil of elliptic curves in $$\mathbb{P}^2.$$ Possible singular fibers of the elliptic fibration $$|-K_\mathcal{E}|$$ were classified by Kodaira. Let's assume that $$\mathcal{E}$$ has exactly one reducible fiber with two irreducible components isomorphic to smooth rational curves.

Then there are two options: these components can intersect in two distinct points ($$I_2$$ fiber) or in one point of multiplicity 2 (type III fiber). Is there a way to distinguish between these two cases looking at ranks of cohomology groups of coherent sheaves on $$\mathcal{E}$$?