It is well-known that there is a universal characterization of blow-ups. However, this one is not so easy to use. According to Hartshorne-Theorem II.8.24. The blow-up of a nonsingular $X$ along a nonsingular subvariety $Y$ have three properties:

1. The blow-up $\tilde{X}$ is nonsingular.
2. The blow-up restricts to the exceptional divisor $\tilde{Y}$ is the projective bundle $P$ of the normal bundle of $Y$ in $X$.
3. The normal sheaf $N_{\tilde{Y}/\tilde{X}}$ corresponds to the $P(-1)$.

My question is that are these three properties enough to characterize the blow-up? Especially, I want to see a counterexample for a non-blowup having property 2. Thank you so much.