# Blow up example

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.

Thanks!!

• 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. – abx May 9 at 14:18
• 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. – Student85 May 10 at 19:25
• 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 ? – Student85 May 10 at 19:32