Let $f: X \longrightarrow Y$ be a morphism from a projective variety $X$ to projective space Y.
How to define the branch locus of $f$?
Where can I find this definition? In general I find only for the case where $X$ and $Y$ are Riemann surfaces. But what interests me is the case above.
Thank you!
Let $A$ be the set of points $y \in Y$ such that $f^{-1}(y)$ consists of n distinct points. The minimal algebraic variety containing the complement of $A$ (i. e. the closure of the complement of $A$) will be denoted by $D$.
The pure part of dimension $d -1$ of $D$ it will be called the branch locus of the rational map $f:X\longrightarrow Y$.
Note: 1) We are assuming here dim $X =$ dim $Y$.
2) The set $ A $ is given by the proposition (3.17), page 46 in David Mumford. Algebraic Geometry 1, Complex Projective Varieties. Springer-Verlag, 1995.