  In this paper we study lumpy black holes with AdS${}_p \times S^q$
asymptotics, where the isometry group coming from the sphere factor is broken
down to SO($q$). Depending on the values of $p$ and $q$, these are solutions to
a certain Supergravity theory with a particular gauge field. We have considered
the values $(p,q) = (5,5)$ and $(p,q) = (4,7)$, corresponding to type IIB
supergravity in ten dimensions and eleven-dimensional supergravity
respectively. These theories presumably contain an infinite spectrum of
families of lumpy black holes, labeled by a harmonic number $\ell$, whose
endpoints in solution space merge with another type of black holes with
different horizon topology. We have numerically constructed the first four
families of lumpy solutions, corresponding to $\ell = 1, 2^+, 2^-$ and $3$. We
show that the geometry of the horizon near the merger is well-described by a
cone over a triple product of spheres, thus extending Kol's local model to the
present asymptotics. Interestingly, the presence of non-trivial fluxes in the
internal sphere implies that the cone is no longer Ricci flat. This conical
manifold accounts for the geometry and the behavior of the physical quantities
of the solutions sufficiently close to the critical point. Additionally, we
show that the vacuum expectation values of the dual scalar operators approach
their critical values with a power law whose exponents are dictated by the
local cone geometry in the bulk.
