Exotic Image Formation in Strong Gravitational Lensing by Clusters of
  Galaxies. I: Cross-Section
  In a recent paper we have discussed the higher order singularities in
gravitational lensing. We have shown that a singularity map, comprising of
$A_3$-lines and unstable (point) singularities ($A_4$ and $D_4$), is a compact
representation of high magnification regions corresponding to a given lens
model for all possible source redshifts. It marks all the optimal locations for
deep surveys in the lens plane. Here we present singularity maps for ten
different cluster lenses selected from the \textit{Hubble Frontier fields}
(HFF) and the \textit{Reionization Lensing Cluster Survey} (RELICS) surveys. We
have identified regions in the lens plane with a high magnification for sources
up to redshift ten. To determine the dependence of unstable (point)
singularities on lens mass model reconstruction techniques, we compared
singularity maps corresponding to the different mass models (provided by
various groups in the HFF survey) for each cluster lens. We find that the
non-parametric (free-form) method of lens mass reconstruction yields the least
number of point singularities. In contrast, mass models reconstructed by
various groups using a parametric approach have a significantly larger number
of point singularities. We also estimate the number of galaxies lying near
these unstable (point) singularities, which can be observed with the
\textit{James Webb Space Telescope} (JWST). We find that we expect to get at
least one hyperbolic umbilic and one swallowtail image formation for a source
at $z > 1$ for every five clusters with JWST. These numbers are much higher
than earlier estimates.
