  The Taubes equation for Abelian Higgs vortices is generalised to five
distinct U(1) vortex equations. These include the Popov and Jackiw--Pi vortex
equations, and two new equations. The Baptista metric, a conformal rescaling of
the background metric by the squared Higgs field, gives insight into these
vortices, and shows that vortices can be interpreted as conical singularities
superposed on the background geometry. When the background has a constant
curvature adapted to the vortex type, then the vortex equation is integrable by
a reduction to Liouville's equation, and the Baptista metric has a constant
curvature too, apart from its conical singularities. The conical geometry is
fairly easy to visualise in some cases.
