  Ising and Potts models can be studied using the Fortuin-Kasteleyn
representation through the Edwards-Sokal coupling. This adapts to the setting
where the models are exposed to an external field of strength $h>0$.
  In this representation, which is also known as the random-cluster model, the
Kert\'{e}sz line separates the two regions of parameter space according to the
existence of an infinite cluster in $\mathbb{Z}^d$. This signifies a geometric
phase transition between the ordered and disordered phases even in cases where
a thermodynamic phase transition does not occur. In this article, we prove
strict monotonicity and continuity of the Kert\'{e}sz line. Furthermore, we
give new rigorous bounds that are asymptotically correct in the limit $h \to 0$
complementing the bounds from the work of Ruiz and Wouts [J. Math. Phys. 49,
053303 (2008)], which were asymptotically correct for $h \to \infty$. Finally,
using a cluster expansion, we investigate the continuity of the Kert\'{e}sz
line phase transition.
