I am working in this case: Let X be a, possibly singular, algebraic variety embedded as a closed subvariety of a manifold(smooth) M$(i : X \longrightarrow M )$. We consider the Blow up $\pi$ of $M$ along to $X$, i.e., $\pi : \widehat{M} \longrightarrow M$ is a proper birational map where $\widehat{M}$ is a manifold such that $\pi^{-1}(X) := X^{'}$ is a divisor with isolated singularities.

$\textbf{Question 1:}$ Is the condition of $X^{'}$ to be a divisor with isolated singularities generic?

$\textbf{Question 2:}$ Can somebody give me, please, an example of this situation, i. e., i need $X, M, \widehat{M}$ and $X^{'}$ in above situation.


  • 2
    $\begingroup$ What do you mean by "generic"? If $X$ is locally a complete intersection in $M$, $X'$ is a projective bundle over $X$, so it has always non-isolated singularities if $X$ is singular. $\endgroup$ – abx May 9 at 14:18
  • $\begingroup$ A property "P" is said generic if its occur many times(in dense sense), i.e., if the set where "P" is valid is a dense set. Thank you. $\endgroup$ – Student85 May 10 at 19:25
  • $\begingroup$ But, if i can consider X be a variety not LCI(locally complete intersecton) as for example a toric variety or a determinantal variety, i still can X with isolated singularities ? $\endgroup$ – Student85 May 10 at 19:32

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