  Motivated by the construction of holographic duals to five-dimensional
superconformal quantum field theories, we obtain global solutions to Type IIB
supergravity invariant under the superalgebra $F(4)$ on a space-time of the
form $AdS_6 \times S^2$ warped over a two-dimensional Riemann surface $\Sigma$.
In earlier work, the general local solutions were expressed in terms of two
locally holomorphic functions $\mathcal A_\pm$ on $\Sigma$ and global solutions
were sketched when $\Sigma$ is a disk. In the present paper, the physical
regularity conditions on the supergravity fields required for global solutions
are implemented on $\mathcal A_\pm$ for arbitrary $\Sigma$. Global solutions
exist only when $\Sigma$ has a non-empty boundary $\partial \Sigma$. The
differentials $\partial \mathcal A_\pm$ are allowed to have poles only on
$\partial \Sigma$ and each pole corresponds to a semi-infinite $(p,q)$
five-brane. The construction for the disk is carried out in detail and the
conditions for the existence of global solutions are articulated for surfaces
with more than one boundary and higher genus.
