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Let $X$ be a projective variety and $Y \subset X$ is a very ample divisor on $X$. Let $Z \subset X$ be a regular subvariety (but $X$ need not be regular along $Z$) of codimension $2$ and $\pi:\tilde{X} \to X$ be the blow-up of $X$ along $Z$. Is there any known condition under which the strict transform of $Y$ in $\tilde{X}$ is ample?

EDIT Also assume that $Z$ is contained in $Y$.

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    $\begingroup$ Even for $X=\mathbb{P}^2$ and $Z$ finite (and reduced), it is a tricky question -- it depends not only on the number of points, but also on their position. I would be very surprised if anything could be said in your general setup. $\endgroup$
    – abx
    Dec 2, 2017 at 7:03
  • $\begingroup$ @abx I do not know if this makes any difference, but in my case $Z$ is irreducible. In particular, translating to your example it is just one point. However, in my case $Z \subset Y$. $\endgroup$
    – Ron
    Dec 2, 2017 at 9:17
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    $\begingroup$ There are blow-ups of P3 along a curve, where the ample cone is extremely complicated, so I agree with abx $\endgroup$
    – byu
    Dec 2, 2017 at 13:13

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