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Claim Let $f:\widetilde X\to X$ be a projective birational morphism between smooth quasi-projective varieties over an algebraically closed field. Suppose that the image $Y$ of the exceptional set of $f$ in $X$ is also smooth. Then $f$ is the blow up of $X$ along $Y$.

Proof By the assumptions $f$ is the blow up of $X$ along an ideal sheaf $\mathscr I\subset \mathscr O_X$. Let $\mathscr J$ be the ideal sheaf of $Y$ and let $\pi: B\to X$ be the blow up of $X$ along $Y$ (i.e., along $\mathscr J$).

Since by assumption the support of $\mathscr O_X/\mathscr I$ is $Y$, it follows that $\mathscr J$ is the radical of $\mathscr I$ or in other words $Y$ is the reduced scheme of the scheme along which one has to blow up $X$ to get $f$. This implies that $\pi$ factors through $f$: $$\pi: B\overset{\sigma}\to \widetilde X\overset{f}\to X,$$ i.e., $\pi=f\circ\sigma$ for an appropriate $\sigma:B\to\widetilde X$.

We may assume that $Y$ is connected and then if it is smooth it has to be irreducible. From the standard facts about blow ups along smooth varieties it follows that the exceptional set of $\pi$ is a single irreducible divisor on $B$ (it is a projective space bundle over the irreducible $Y$).

Now observe that if $\widetilde X$ is smooth (actually already if it is normal and has $\mathbb Q$-factorial singularities), then $\sigma$ is either an isomorphism or it has an exceptional divisor. The same holds for $f$, but there is only one divisor that is exceptional for $\pi$. Therefore either $\sigma$ or $f$ has to be an isomorphism. $\square$

Remark the proof suggests that even assuming that $\widetilde X$ is normal and has $\mathbb Q$-factorial singularities is enough for this characterization. Accordingly in the example below, when the ideal $(x^2,y^2)$ is blown up on the plane, then $\widetilde X$ is not normal and the blow up of the reduced point is the normalization (which happens to be smooth) of the pinch point surface $x^2z=y^2$.

Example (property 1 fails, but property 2 is satisfied)

Look for $f$ as the blow up of an ideal sheaf $\mathscr I$, so $\widetilde X=\mathrm{Proj}_X(\oplus_d \mathscr I^d)$. Then the pre-image of the subscheme $Z\subset X$ defined by the ideal $\mathscr I$ is given by $\widetilde Z=\mathrm{Proj}_Y(\oplus_d \mathscr{I^d/I^{d+1}})$. Now if $X$ is Cohen-Macaulay and $Z$ is a complete intersection in $X$, (i.e., $\mathscr I$ is generated by a regular sequence), then $\mathscr{I/I^2}$ is locally free and $\mathscr{I^d/I^{d+1}}\simeq \mathrm{Sym}^d(\mathscr{I/I^2})$ and hence $\widetilde Z\simeq \mathbb P(\mathscr{I/I^2})$.

Property #3 is kind of a red herring. The $(-1)$-twist is almost automatic, it comes from the construction of the blow up of $\mathscr I$.

Finally, here is a simple concrete example: Let $X$ be a plane (or any smooth surface) and $\mathscr I=(x^2,y^2)$ where $x,y$ are local coordinates at a point. The blow up will be the surface with a pinch point (locally around the interesting singularity defined by $x^2z=y^2$) with the singular line contracted to a point. I think it is relatively easy to check that this satisfies properties #2 and #3.

To round things up Mike Roth in the comments below gives a nice example of a blow up along a non-smooth subvariety such that the resulting variety is actually smooth.

Example (property 1 fails, but property 2 is satisfied)

Look for $f$ as the blow up of an ideal sheaf $\mathscr I$, so $\widetilde X=\mathrm{Proj}_X(\oplus_d \mathscr I^d)$. Then the pre-image of the subscheme $Z\subset X$ defined by the ideal $\mathscr I$ is given by $\widetilde Z=\mathrm{Proj}_Y(\oplus_d \mathscr{I^d/I^{d+1}})$. Now if $X$ is Cohen-Macaulay and $Z$ is a complete intersection in $X$, (i.e., $\mathscr I$ is generated by a regular sequence), then $\mathscr{I/I^2}$ is locally free and $\mathscr{I^d/I^{d+1}}\simeq \mathrm{Sym}^d(\mathscr{I/I^2})$ and hence $\widetilde Z\simeq \mathbb P(\mathscr{I/I^2})$.

Property #3 is kind of a red herring. The $(-1)$-twist is almost automatic, it comes from the construction of the blow up of $\mathscr I$.

Finally, here is a simple concrete example: Let $X$ be a plane (or any smooth surface) and $\mathscr I=(x^2,y^2)$ where $x,y$ are local coordinates at a point. The blow up will be the surface with a pinch point (locally around the interesting singularity defined by $x^2z=y^2$) with the singular line contracted to a point. I think it is relatively easy to check that this satisfies properties #2 and #3.

To round things up Mike Roth in the comments below gives a nice example of a blow up along a non-smooth subvariety such that the resulting variety is actually smooth.

EDIT: This was an answer when the last line of the question said "Especially, I want to see an example for a non-blowup having property 2."

I think you should make your question more precise. If $X$ is quasi-projective, and $f:\widetilde X\to X$ is a projective birational morphism, then $f$ is a blow-up of an appropriate ideal sheaf. I am guessing this is not what you're after. (If you allow blow ups of fractional ideals, then you can drop even the quasi-projective assumption. Not assuming that $f$ is projective would make the question less interesting).

Also the way you phrased your question, I think there is a simple example: take any blow ups and then choose a subscheme of $X$ with the same support, but different scheme structure than the subscheme that was blown up.

Anyway, I will assume that you want an example that's not a blow up centered at a smooth subvariety.

I think it is possible to give examples satisfying property #2 relatively easily.

Let's look for $f$ as the blow up of an ideal sheaf $\mathscr I$. So $\widetilde X=\mathrm{Proj}_X(\oplus_d \mathscr I^d)$. Then the pre-image of the subvariety $Y\subset X$ defined by the ideal $\mathscr I$ is given by $\widetilde Y=\mathrm{Proj}_Y(\oplus_d \mathscr{I^d/I^{d+1}})$. Now if $X$ is Cohen-Macaulay and $Y$ is a complete intersection in $X$, (i.e., $\mathscr I$ is generated by a regular sequence), then $\mathscr{I/I^2}$ is locally free and $\mathscr{I^d/I^{d+1}}\simeq \mathrm{Sym}^d(\mathscr{I/I^2})$ and hence $\widetilde Y\simeq \mathbb P(\mathscr{I/I^2})$.

Property #3 is kind of a red herring. The $(-1)$-twist is almost automatic, it comes from the construction of the blow up of $\mathscr I$.

My example is not likely to satisfy property #1, that is $\widetilde X$ is probably always singular if $Y$ is.

As far as finding an example that satisfies all 3 properties, I would not hold my hopes very high. If $\dim X=2$, then these properties just mean that we have a $(-1)$-curve and we know that that is always the blow up of a smooth point. In higher dimensions I would start by taking general hyperplane cuts to reduce to the case when the image is a point. Then Property #2 says that the preimage is a $\mathbb P^n$ with normal bundle $\mathscr O_{\mathbb P^n}(-1)$ so there is a good chance that one can prove that it is indeed a blow up. Then one would get (probably) that the original map is a blow up at general points and then possibly one can prove that then it has to be a blow up everywhere, but I am not entirely sure about that.

EDIT Mike Roth in the comments below gives a nice example of a blow up along a non-smooth subvariety such that the resulting variety is actually smooth. I don't think this satisfies the other conditions, but it might suggest that proving that those three conditions indeed characterize blow ups along smooth subvarieties is probably not easy.

Claim Let $f:\widetilde X\to X$ be a projective birational morphism between smooth quasi-projective varieties over an algebraically closed field. Suppose that the image $Y$ of the exceptional set of $f$ in $X$ is also smooth. Then $f$ is the blow up of $X$ along $Y$.

Proof By the assumptions $f$ is the blow up of $X$ along an ideal sheaf $\mathscr I\subset \mathscr O_X$. Let $\mathscr J$ be the ideal sheaf of $Y$ and let $\pi: B\to X$ be the blow up of $X$ along $Y$ (i.e., along $\mathscr J$).

Since by assumption the support of $\mathscr O_X/\mathscr I$ is $Y$, it follows that $\mathscr J$ is the radical of $\mathscr I$ or in other words $Y$ is the reduced scheme of the scheme along which one has to blow up $X$ to get $f$. This implies that $\pi$ factors through $f$: $$\pi: B\overset{\sigma}\to \widetilde X\overset{f}\to X,$$ i.e., $\pi=f\circ\sigma$ for an appropriate $\sigma:B\to\widetilde X$.

We may assume that $Y$ is connected and then if it is smooth it has to be irreducible. From the standard facts about blow ups along smooth varieties it follows that the exceptional set of $\pi$ is a single irreducible divisor on $B$ (it is a projective space bundle over the irreducible $Y$).

Now observe that if $\widetilde X$ is smooth (actually already if it is normal and has $\mathbb Q$-factorial singularities), then $\sigma$ is either an isomorphism or it has an exceptional divisor. The same holds for $f$, but there is only one divisor that is exceptional for $\pi$. Therefore either $\sigma$ or $f$ has to be an isomorphism. $\square$

Remark the proof suggests that even assuming that $\widetilde X$ is normal and has $\mathbb Q$-factorial singularities is enough for this characterization. Accordingly in the example below, when the ideal $(x^2,y^2)$ is blown up on the plane, then $\widetilde X$ is not normal and the blow up of the reduced point is the normalization (which happens to be smooth) of the pinch point surface $x^2z=y^2$.

Example (property 1 fails, but property 2 is satisfied)

Look for $f$ as the blow up of an ideal sheaf $\mathscr I$, so $\widetilde X=\mathrm{Proj}_X(\oplus_d \mathscr I^d)$. Then the pre-image of the subscheme $Z\subset X$ defined by the ideal $\mathscr I$ is given by $\widetilde Z=\mathrm{Proj}_Y(\oplus_d \mathscr{I^d/I^{d+1}})$. Now if $X$ is Cohen-Macaulay and $Z$ is a complete intersection in $X$, (i.e., $\mathscr I$ is generated by a regular sequence), then $\mathscr{I/I^2}$ is locally free and $\mathscr{I^d/I^{d+1}}\simeq \mathrm{Sym}^d(\mathscr{I/I^2})$ and hence $\widetilde Z\simeq \mathbb P(\mathscr{I/I^2})$.

Property #3 is kind of a red herring. The $(-1)$-twist is almost automatic, it comes from the construction of the blow up of $\mathscr I$.

To round things up Mike Roth in the comments below gives a nice example of a blow up along a non-smooth subvariety such that the resulting variety is actually smooth.

I think you should make your question more precise. If $X$ is quasi-projective, and $f:\widetilde X\to X$ is a projective birational morphism, then $f$ is a blow-up of an appropriate ideal sheaf. I am guessing this is not what you're after. (If you allow blow ups of fractional ideals, then you can drop even the quasi-projective assumption. Not assuming that $f$ is projective would make the question less interesting).

Also the way you phrased your question, I think there is a simple example: take any blow ups and then choose a subscheme of $X$ with the same support, but different scheme structure than the subscheme that was blown up.

Anyway, I will assume that you want an example that's not a blow up centered at a smooth subvariety.

I think it is possible to give examples satisfying property #2 relatively easily.

Let's look for $f$ as the blow up of an ideal sheaf $\mathscr I$. So $\widetilde X=\mathrm{Proj}_X(\oplus_d \mathscr I^d)$. Then the pre-image of the subvariety $Y\subset X$ defined by the ideal $\mathscr I$ is given by $\widetilde Y=\mathrm{Proj}_Y(\oplus_d \mathscr{I^d/I^{d+1}})$. Now if $X$ is Cohen-Macaulay and $Y$ is a complete intersection in $X$, (i.e., $\mathscr I$ is generated by a regular sequence), then $\mathscr{I/I^2}$ is locally free and $\mathscr{I^d/I^{d+1}}\simeq \mathrm{Sym}^d(\mathscr{I/I^2})$ and hence $\widetilde Y\simeq \mathbb P(\mathscr{I/I^2})$.

Property #3 is kind of a red herring. The $(-1)$-twist is almost automatic, it comes from the construction of the blow up of $\mathscr I$.

My example is not likely to satisfy property #1, that is $\widetilde X$ is probably always singular if $Y$ is.

Finally, here is a simple concrete example: Let $X$ be a plane (or any smooth surface) and $\mathscr I=(x^2,y^2)$ where $x,y$ are local coordinates at a point. The blow up will be the surface with a pinch point (locally around the interesting singularity defined by $x^2z=y^2$) with the singular line contracted to a point. I think it is relatively easy to check that this satisfies properties #2 and #3.

As far as finding an example that satisfies all 3 properties, I would not hold my hopes very high. If $\dim X=2$, then these properties just mean that we have a $(-1)$-curve and we know that that is always the blow up of a smooth point. In higher dimensions I would start by taking general hyperplane cuts to reduce to the case when the image is a point. Then Property #2 says that the preimage is a $\mathbb P^n$ with normal bundle $\mathscr O_{\mathbb P^n}(-1)$ so there is a good chance that one can prove that it is indeed a blow up. Then one would get (probably) that the original map is a blow up at general points and then possibly one can prove that then it has to be a blow up everywhere, but I am not entirely sure about that.

EDIT Mike Roth in the comments below gives a nice example of a blow up along a non-smooth subvariety such that the resulting variety is actually smooth. I don't think this satisfies the other conditions, but it might suggest that proving that those three conditions indeed characterize blow ups along smooth subvarieties is probably not easy.

EDIT: This was an answer when the last line of the question said "Especially, I want to see an example for a non-blowup having property 2."

I think you should make your question more precise. If $X$ is quasi-projective, and $f:\widetilde X\to X$ is a projective birational morphism, then $f$ is a blow-up of an appropriate ideal sheaf. I am guessing this is not what you're after. (If you allow blow ups of fractional ideals, then you can drop even the quasi-projective assumption. Not assuming that $f$ is projective would make the question less interesting).

Also the way you phrased your question, I think there is a simple example: take any blow ups and then choose a subscheme of $X$ with the same support, but different scheme structure than the subscheme that was blown up.

Anyway, I will assume that you want an example that's not a blow up centered at a smooth subvariety.

I think it is possible to give examples satisfying property #2 relatively easily.

Let's look for $f$ as the blow up of an ideal sheaf $\mathscr I$. So $\widetilde X=\mathrm{Proj}_X(\oplus_d \mathscr I^d)$. Then the pre-image of the subvariety $Y\subset X$ defined by the ideal $\mathscr I$ is given by $\widetilde Y=\mathrm{Proj}_Y(\oplus_d \mathscr{I^d/I^{d+1}})$. Now if $X$ is Cohen-Macaulay and $Y$ is a complete intersection in $X$, (i.e., $\mathscr I$ is generated by a regular sequence), then $\mathscr{I/I^2}$ is locally free and $\mathscr{I^d/I^{d+1}}\simeq \mathrm{Sym}^d(\mathscr{I/I^2})$ and hence $\widetilde Y\simeq \mathbb P(\mathscr{I/I^2})$.

Property #3 is kind of a red herring. The $(-1)$-twist is almost automatic, it comes from the construction of the blow up of $\mathscr I$.

My example is not likely to satisfy property #1, that is $\widetilde X$ is probably always singular if $Y$ is.

Finally, here is a simple concrete example: Let $X$ be a plane (or any smooth surface) and $\mathscr I=(x^2,y^2)$ where $x,y$ are local coordinates at a point. The blow up will be the surface with a pinch point (locally around the interesting singularity defined by $x^2z=y^2$) with the singular line contracted to a point. I think it is relatively easy to check that this satisfies properties #2 and #3.

As far as finding an example that satisfies all 3 properties, I would not hold my hopes very high. If $\dim X=2$, then these properties just mean that we have a $(-1)$-curve and we know that that is always the blow up of a smooth point. In higher dimensions I would start by taking general hyperplane cuts to reduce to the case when the image is a point. Then Property #2 says that the preimage is a $\mathbb P^n$ with normal bundle $\mathscr O_{\mathbb P^n}(-1)$ so there is a good chance that one can prove that it is indeed a blow up. Then one would get (probably) that the original map is a blow up at general points and then possibly one can prove that then it has to be a blow up everywhere, but I am not entirely sure about that.

EDIT Mike Roth in the comments below gives a nice example of a blow up along a non-smooth subvariety such that the resulting variety is actually smooth. I don't think this satisfies the other conditions, but it might suggest that proving that those three conditions indeed characterize blow ups along smooth subvarieties is probably not easy.