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Let $Z\subseteq Sing(X)\subset M$, where all objects are complex reduced and projective, moreover $Z$ and $M$ are smooth. Consider the strict transform $\tilde{X}\subset Bl_Z M$.

  1. If all the singularities of $X$ are of hypersurface type then so are all the singularities of $\tilde{X}$. I guess this is true even if the (smooth) center of blowup lies in $X$ but is not contained in $Sing(X)$. Is the same true for locally complete intersections (at least for the case $Z\subseteq Sing(X)$)? Is there some even bigger class of singularities that is preserved under blowups with smooth centers? (something more restrictive than Cohen-Macaulay) Related questions have been asked here (1, 2) but I can't find an answer there :(

  2. Let $(X,0)=\{f=0\}\subset(\Bbb C^N,0)$ be an isolated hypersurface singularity of multiplicity $p$. The strict transform $\tilde{X}\subset Bl_0(\Bbb C^N,0)$ admits a very specific smoothing that comes from downstairs: $(X_{\epsilon},0)=\{f+\epsilon (x^p_1+..+x^p_n)=0\}$. I.e. the (flat) equi-multiple family $(X_{\epsilon},0)$ singularities that possesses simultaneous strict transform. "Some sort of smoothing with descent". I guess the same is true for locally complete intersections, right? Is it true in some more general situation?

  3. Suppose $X\subset M$ is a hypersurface, the zero locus of a section $s\in\Gamma(\mathcal{L}_M)$. Let $p=mult_Z X$ be the generic multiplicity. Then the strict transform $\tilde{X}\subset Bl_ZM$ is the zero locus of a section of the bundle $\pi^*\mathcal{L}_M(-pE)$. One can write a similar formula for $X\subset M$ a complete intersection. What about a more general case?

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