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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 assume we have a projective birational morphism $f: \tilde{X}\to X$ with exceptional locus $Y$ on $X$. We assume both $X$ and $Y$ are nonsingular as before. Let $\tilde{Y}$ be the inverse image sheaf of $Y$. If $f$ satisfies the above three properties, must it be the blow-up of $X$ along $Y$? Especially, I want to see an example for a non-blowup having property 1,2. Thank you so much.

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 assume we have a projective birational morphism $f: \tilde{X}\to X$ with exceptional locus $Y$ on $X$. We assume both $X$ and $Y$ are nonsingular as before (even projective). Let $\tilde{Y}$ be the inverse image sheaf of $Y$. If $f$ satisfies the above three properties, must it be the blow-up of $X$ along $Y$? Especially, I want to see an example for a non-blowup having property 1,2. Thank you so much.

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 assume we have a projective birational morphism $f: \tilde{X}\to X$ with exceptional locus $Y$ on $X$. We assume both $X$ and $Y$ are nonsingular as before. Let $\tilde{Y}$ be the inverse image sheaf of $Y$. If $f$ satisfies the above three properties, must it be the blow-up of $X$ along $Y$? Especially, I want to see a counterexample for a non-blowup having property 1,2. Thank you so much.

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 assume we have a projective birational morphism $f: \tilde{X}\to X$ with exceptional locus $Y$ on $X$. We assume both $X$ and $Y$ are nonsingular as before. Let $\tilde{Y}$ be the inverse image sheaf of $Y$. If $f$ satisfies the above three properties, must it be the blow-up of $X$ along $Y$? Especially, I want to see an example for a non-blowup having property 1,2. Thank you so much.

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 assume we have a birational morphism $f: \tilde{X}\to X$ with exceptional locus $Y$ on $X$. Let $\tilde{Y}$ be the inverse image sheaf of $Y$. If $f$ satisfies the above three properties, must it be the blow-up of $X$ along $Y$? Especially, I want to see a counterexample for a non-blowup having property 2. Thank you so much.

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 assume we have a projective birational morphism $f: \tilde{X}\to X$ with exceptional locus $Y$ on $X$. We assume both $X$ and $Y$ are nonsingular as before. Let $\tilde{Y}$ be the inverse image sheaf of $Y$. If $f$ satisfies the above three properties, must it be the blow-up of $X$ along $Y$? Especially, I want to see a counterexample for a non-blowup having property 1,2. Thank you so much.