A branched covering map is a map 
that is an ordinary covering
map away from its branch points. Its branch locus
is the subset of 
over which 
fails to be an ordinary covering
map. Equivalently, it is the image of the points in
at which 
is locally ramified. Here
"locally" means after restricting 
to sufficiently small neighborhoods
of the point and its image.
Branch Locus
See also
Branch Point, Covering Map, Image, RamificationExplore with Wolfram|Alpha
References
Miranda, R. Algebraic Curves and Riemann Surfaces. Providence, RI: Amer. Math. Soc., 1995.Cite this as:
Weisstein, Eric W. "Branch Locus." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BranchLocus.html