For a smooth map $$f:N\longrightarrow M$$ of manifolds, a point $p$ in $N$ is a critical point of $f$ if the differential $$f_{*,p}:T_{p}N\longrightarrow T_{f(p)}M$$ fails to be surjective.

My problem:

I'm reading some points about map $f:X\longrightarrow Y$ morphism of varieties (Singularities of a Map, Étale maps, degree of maps ...) and I found expressions like:

1) "$x \in X$ a point where the map $f$ is singular."

Question: Is this the same as saying that $ x $ is a critical point of $ f $?

2) "$x \in X$ a point where $df$ is degenerate." $\\$

Quetion: Is this the same as saying that $ x $ is a critical point of $ f $?

Thank You!

  • 2
    $\begingroup$ This ultimately depends on your definitions of singular and degenerate. $\endgroup$ – B. Mehta Apr 14 '18 at 23:19
  • $\begingroup$ That is the problem. I want to know if expressions 1) and 2) are used in what sense? They are not my definitions. I have found in books and would like to know if they can be used as in the definition of critical point of $f$. $\endgroup$ – Manoel Apr 14 '18 at 23:27
  • 1
    $\begingroup$ Essentially, the answer is yes, but you should be careful here because the definitional issues matter. $\endgroup$ – KReiser Apr 16 '18 at 1:28
  • $\begingroup$ I'm being careful. I faced some similar difficulty with the expression "general position", "general element". $\endgroup$ – Manoel Apr 16 '18 at 1:39
  • 1
    $\begingroup$ If $x ,f(x)$ are smooth points of $X,Y$ respectively then I guess so, but if not then the differential geometry perspective doesn't immediately translate, so I guess one has to be more algebraic $\endgroup$ – Nick L Apr 16 '18 at 11:02

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