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Let $X$ be a smooth affine variety over a field, $Z\subset Y\subset X$ are closed (reduced) subvarieties. What are the possible ways to verify whether $Y$ is singular at $Z$ i.e. whether $Z$ is contained in the singularity locus of $Y$? I would prefer some conditions in terms of ideals that determine $Y$ and $Z$ in $X$; yet other criteria could also be useful.

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    $\begingroup$ Can't you compute the singular locus using the Jacobian criterion? $\endgroup$
    – Thomas Kahle
    Commented Apr 6, 2013 at 7:22
  • $\begingroup$ I would also like to know other possibilities. $\endgroup$
    – Mikhail Bondarko
    Commented Apr 6, 2013 at 7:23
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    $\begingroup$ You can always look at the dimension of the tangent space at the generic point of $Z$ of course too (ie, use the definition of regular local ring). This can be done in terms of ideals of $Y$ and $Z$ certainly (ie localize at the prime ideal defining $Z$ in $O_X/I_Y$ and compute the dimension of $I_Z/I_Z^2$). If the field is of characteristic $p$, you can look at whether $F_* O_Y$ is a flat $O_Y$-module along $Z$ (Kunz's criterion). $\endgroup$
    – Karl Schwede
    Commented Apr 6, 2013 at 11:18

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