 Total ramification of a holomorphic map

Count ramification points upstairs in $X$, weighted by $e_p-1$, rather than counting distinct branch values of a holomorphic map downstairs in $Y$. Grouping by fibres givesThe cardinality counts distinct points; the degree of a holomorphic map counts them with their analytic ramification indices. Thus each summand is the number of missing distinct preimages in that fibre.^[ref]

The phrase branch point varies across conventions: to keep the two surfaces distinct, use ramification point for $p\in X$ and branch value of a holomorphic map for $f(p)\in Y$. The existing name branch point of a holomorphic map here denotes the latter. Several ramification points can share one branch value of a holomorphic map.