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 $\mathcal{E}$?


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