# what does “The pure part of dimension $d -1$ of $D$” mean?

In A. Andreotti On to Theorem of Torelli American Journal of Mathematics Vol. 80, No. 4, pp. 801-828, it is written:

Let $f: X \longrightarrow P$ be a rational map of a projective irreducible variety of dimension $d$ on a projective space $P_d$.

Let $A$ be the set of points $y \in P$ 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 $D$ has an intrinsic meaning in terms of the extension $k(P)\subset k(X)$ and the chosen model $P$ of $k(P)$. It will be called the branch locus of the rational map $f: X \longrightarrow P$.

My question is: what does "The pure part of dimension $d -1$ of $D$" mean?

• It means the portion of $D$ consisting only of the points at which $D$ has dimension $d-1$. – KReiser Apr 7 '18 at 0:22
• What does it mean $D$ have a dimension $d-1$ at a point? – Manoel Apr 7 '18 at 2:14
• stacks.math.columbia.edu/tag/04MS – KReiser Apr 7 '18 at 2:27
• Your answer is good for me, because Definition 27.10.1 is for a scheme. And I already asked in mathoverflow.net/questions/296797/… how I could see a relationship between schemes and projective varieties – Manoel Apr 7 '18 at 2:38
• Yes, certainly. Projective varieties embed fully faithfully into the category of all schemes. – KReiser Apr 7 '18 at 2:55