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Let $f:V\to W$ be a morphism between varieties, with $\dim \overline{f(V)} = \dim V$. What do you call the closed proper subvariety $S$ of $V$ consisting of points $x$ such that $\textrm{dim} f^{-1}(f(x))>0$?

("Singular points of $f$" doesn't seem quite right.)

More generally, if $\dim \overline{f(V)}$ is not necessarily $\dim V$: is there a name for the closed proper subvariety of $V$ consisting of points $x$ such that $\textrm{dim} f^{-1}(f(x)) > r$, where $r = \dim V - \dim \overline{f(V)}$?

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    $\begingroup$ It is the exceptional locus of the connected part of the Stein factorization. $\endgroup$
    – Jason Starr
    Commented Mar 2, 2023 at 11:56
  • $\begingroup$ @JasonStarr : meaning what exactly, in the notation in en.wikipedia.org/wiki/Stein_factorization ? $\endgroup$
    – H A Helfgott
    Commented Mar 2, 2023 at 13:23
  • $\begingroup$ In that notation, the morphism $f’$ is birational, and your locus is the exceptional locus of this birational morphism. $\endgroup$
    – Jason Starr
    Commented Mar 2, 2023 at 13:49
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    $\begingroup$ What one gets then is not all $x$ such that $\textrm{dim} f^{-1}(f(x))>0$, but only those sitting in components of $f^{-1}(f(x))$ of positive dimension, right? $\endgroup$
    – H A Helfgott
    Commented Mar 2, 2023 at 15:14
  • $\begingroup$ Does one also capture the case $r>0$ in exactly the same way using the Stein factorization? $\endgroup$
    – H A Helfgott
    Commented Mar 3, 2023 at 10:22

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