  This article consists of a detailed geometric study of the one-dimensional
vorticity model equation $$\omega_{t} + u\omega_{x} + 2\omega u_{x} = 0, \qquad
\omega = H u_{x}, \qquad t\in\mathbb{R},\; x\in S^{1}\,,$$ which is a
particular case of the generalized Constantin-Lax-Majda equation. Wunsch showed
that this equation is the Euler-Arnold equation on $\operatorname{Diff}(S^{1})$
when the latter is endowed with the right-invariant homogeneous
$\dot{H}^{1/2}$-metric. In this article we prove that the exponential map of
this Riemannian metric is not Fredholm and that the sectional curvature is
locally unbounded. Furthermore, we prove a Beale-Kato-Majda-type blow-up
criterion, which we then use to demonstrate a link to our non-Fredholmness
result. Finally, we extend a blow-up result of Castro-C\'ordoba to the periodic
case and to a much wider class of initial conditions, using a new
generalization of an inequality for Hilbert transforms due to
C\'ordoba-C\'ordoba.
