Matter from multiply enhanced singularities in F-theory

We investigate the geometrical structure of multiply enhanced codimension-two singularities in the $SU(5)$ model of six-dimensional F-theory, where the rank of the singularity increases by two or more. We perform blow-up processes for the enhancement $SU(5)\rightarrow G'$, where $G'=E_6$, $E_7$ or $E_8$, to examine whether a sufficient set of exceptional curves emerge that can explain the charged matter generation predicted from anomaly cancelation. We first apply one of the six Esole-Yau small resolutions to the multiply enhanced singularities, but it turns out that the proper transform of the threefold equation does not reflect changes in the singularity or how the generic codimension-two singularities gather there. We then use a(n) (apparently) different way of small resolutions than Esole-Yau to find that, except for the cases of $G'= E_6$ and special cases of $E_7$, either (1) the resolution only yields exceptional curves that are insufficient to cancel the anomaly, or (2) there arises a type of singularity that is neither a conifold nor a generalized conifold singularity. Finally, we revisit the Esole-Yau small resolution and show that the change of the way of small resolutions amounts to simply exchanging the proper transform and the constraint condition, and under this exchange the two ways of small resolutions are completely equivalent.