Anomaly-free multiple singularity enhancement in F-theory

We study global Calabi-Yau realizations of multiple singularity enhancement relevant for family-unification model building in F-theory. We examine the conditions under which the generation of extra chiral matter at multiple singularities on 7-branes in six-dimensional F-theory can be consistent with anomaly cancellation. It is shown that the generation of extra matter is consistent only if it is accompanied by simultaneous degenerations of loci of the leading polynomial of the discriminant so that the total number of chiral matter does not change. We also show that the number of singlets expected to arise matches the decrease of the complex structure moduli for the restricted geometry.