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 $Y$.