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A general fact about the singular locus $Sing(X)$ of a variety $X$ (analytic or projective) is that they form a subvariety of the oringinal variety $X$. And we know that the boundary of a manifold have no boundary itself. My simple question is that

Is $Sing(X)$ (as a variety itself) a smooth variety ?

Intuitively, I can't imagine a picture such that the answer is no.

If the answer is no, another question is that does the singular locus $Sing(X)$ necessarily have dimension less than that of $X$ ?

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  • $\begingroup$ If $X$ is the region in $\mathbb{R}^3$ in which all coordinates are non-negative, then the singular set of $X$ is the union of the three non-negative axes, which is not smooth. $\endgroup$
    – Eduardo Longa
    Jul 29, 2016 at 2:37
  • $\begingroup$ @EduardoLonga That is not an algebraic variety, and I'm not sure what object you're trying to describe (and hence what sense in which that's the singular set). $\endgroup$
    – user98602
    Jul 29, 2016 at 2:41
  • $\begingroup$ I was afraid not.. $\endgroup$
    – Eduardo Longa
    Jul 29, 2016 at 2:42
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    $\begingroup$ Take the union of three coordinate planes in ${\mathbb C}^3$. If you want an irreducible example, take the quotient of the previous example by the action of the permutation group $S_3$. $\endgroup$
    – Moishe Kohan
    Jul 29, 2016 at 3:24
  • $\begingroup$ @studiosus , thanks for your example! $\endgroup$
    – Wei Xia
    Jul 29, 2016 at 4:49

1 Answer 1

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What is true is that the singular locus of a variety has lower dimension. It can be singular, and its singular locus singular, and the singular locus of the latter singular, and so on.

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  • $\begingroup$ So start from any X, eventually (no more than dim(X) steps) we will get a smooth variety, possibly points? $\endgroup$
    – Wei Xia
    Jul 29, 2016 at 4:46
  • $\begingroup$ Exactly. ${}{}{}$ $\endgroup$
    – Mariano Suárez-Álvarez
    Jul 29, 2016 at 5:19
  • $\begingroup$ If $X$ is a given variety, then is there a variety $Y$ whose singular locus is $X$? $\endgroup$
    – Brian Fitzpatrick
    Jul 29, 2016 at 6:23
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    $\begingroup$ @BrianFitzpatrick, ask a new question. Asking new things in comments is not a good idea! $\endgroup$
    – Mariano Suárez-Álvarez
    Jul 29, 2016 at 6:24
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    $\begingroup$ @MarianoSuárez-Alvarez I was mainly curious if you knew if the answer was yes or no. $\endgroup$
    – Brian Fitzpatrick
    Jul 29, 2016 at 6:25

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