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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?

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    $\begingroup$ It means the portion of $D$ consisting only of the points at which $D$ has dimension $d-1$. $\endgroup$ – KReiser Apr 7 '18 at 0:22
  • $\begingroup$ What does it mean $ D $ have a dimension $ d-1 $ at a point? $\endgroup$ – Manoel Apr 7 '18 at 2:14
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    $\begingroup$ stacks.math.columbia.edu/tag/04MS $\endgroup$ – KReiser Apr 7 '18 at 2:27
  • $\begingroup$ 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 $\endgroup$ – Manoel Apr 7 '18 at 2:38
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    $\begingroup$ Yes, certainly. Projective varieties embed fully faithfully into the category of all schemes. $\endgroup$ – KReiser Apr 7 '18 at 2:55

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