Lemma 54.16.1. Let $X$ be a Noetherian scheme. Let $E \subset X$ be an exceptional curve of the first kind. If a contraction $X \to X'$ of $E$ exists, then it has the following universal property: for every morphism $\varphi : X \to Y$ such that $\varphi (E)$ is a point, there is a unique factorization $X \to X' \to Y$ of $\varphi $.

## 54.16 Contracting exceptional curves

Let $X$ be a Noetherian scheme. Let $E \subset X$ be a closed subscheme with the following properties

$E$ is an effective Cartier divisor on $X$,

there exists a field $k$ and an isomorphism $\mathbf{P}^1_ k \to E$ of schemes,

the normal sheaf $\mathcal{N}_{E/X}$ pulls back to $\mathcal{O}_{\mathbf{P}^1}(-1)$.

Such a closed subscheme is called an *exceptional curve of the first kind*.

Let $X'$ be a Noetherian scheme and let $x \in X'$ be a closed point such that $\mathcal{O}_{X', x}$ is regular of dimension $2$. Let $b : X \to X'$ be the blowing up of $X'$ at $x$. In this case the exceptional fibre $E \subset X$ is an exceptional curve of the first kind. This follows from Lemma 54.3.1.

Question: Is every exceptional curve of the first kind obtained as the fibre of a blowing up as above? In other words, does there always exist a proper morphism of schemes $X \to X'$ such that $E$ maps to a closed point $x \in X'$, such that $\mathcal{O}_{X', x}$ is regular of dimension $2$, and such that $X$ is the blowing up of $X'$ at $x$. If true we say *there exists a contraction of $E$*.

**Proof.**
Let $b : X \to X'$ be a contraction of $E$. As a topological space $X'$ is the quotient of $X$ by the relation identifying all points of $E$ to one point. Namely, $b$ is proper (Divisors, Lemma 31.32.13 and Morphisms, Lemma 29.43.5) and surjective, hence defines a submersive map of topological spaces (Topology, Lemma 5.6.5). On the other hand, the canonical map $\mathcal{O}_{X'} \to b_*\mathcal{O}_ X$ is an isomorphism. Namely, this is clear over the complement of the image point $x \in X'$ of $E$ and on stalks at $x$ the map is an isomorphism by part (4) of Lemma 54.3.4. Thus the pair $(X', \mathcal{O}_{X'})$ is constructed from $X$ by taking the quotient as a topological space and endowing this with $b_*\mathcal{O}_ X$ as structure sheaf.

Given $\varphi $ we can let $\varphi ' : X' \to Y$ be the unique map of topological spaces such that $\varphi = \varphi ' \circ b$. Then the map

is adjoint to a map

Then $(\varphi ', (\varphi ')^\sharp )$ is a morphism of ringed spaces from $X'$ to $Y$ such that we get the desired factorization. Since $\varphi $ is a morphism of locally ringed spaces, it follows that $\varphi '$ is too. Namely, the only thing to check is that the map $\mathcal{O}_{Y, y} \to \mathcal{O}_{X', x}$ is local, where $y \in Y$ is the image of $E$ under $\varphi $. This is true because an element $f \in \mathfrak m_ y$ pulls back to a function on $X$ which is zero in every point of $E$ hence the pull back of $f$ to $X'$ is a function defined on a neighbourhood of $x$ in $X'$ with the same property. Then it is clear that this function must vanish at $x$ as desired. $\square$

Lemma 54.16.2. Let $X$ be a Noetherian scheme. Let $E \subset X$ be an exceptional curve of the first kind. If there exists a contraction of $E$, then it is unique up to unique isomorphism.

**Proof.**
This is immediate from the universal property of Lemma 54.16.1.
$\square$

Lemma 54.16.3. Let $X$ be a Noetherian scheme. Let $E \subset X$ be an exceptional curve of the first kind. Let $E_ n = nE$ and denote $\mathcal{O}_ n$ its structure sheaf. Then

is a complete local Noetherian regular local ring of dimension $2$ and $\mathop{\mathrm{Ker}}(A \to H^0(E_ n, \mathcal{O}_ n))$ is the $n$th power of its maximal ideal.

**Proof.**
Recall that there exists an isomorphism $\mathbf{P}^1_ k \to E$ such that the normal sheaf of $E$ in $X$ pulls back to $\mathcal{O}(-1)$. Then $H^0(E, \mathcal{O}_ E) = k$. We will denote $\mathcal{O}_ n(iE)$ the restriction of the invertible sheaf $\mathcal{O}_ X(iE)$ to $E_ n$ for all $n \geq 1$ and $i \in \mathbf{Z}$. Recall that $\mathcal{O}_ X(-nE)$ is the ideal sheaf of $E_ n$. Hence for $d \geq 0$ we obtain a short exact sequence

Since $\mathcal{O}_ E(-(d + n)E) = \mathcal{O}_{\mathbf{P}^1_ k}(d + n)$ the first cohomology group vanishes for all $d \geq 0$ and $n \geq 1$. We conclude that the transition maps of the system $H^0(E_ n, \mathcal{O}_ n(-dE))$ are surjective. For $d = 0$ we get an inverse system of surjections of rings such that the kernel of each transition map is a nilpotent ideal. Hence $A = \mathop{\mathrm{lim}}\nolimits H^0(E_ n, \mathcal{O}_ n)$ is a local ring with residue field $k$ and maximal ideal

Pick $x, y$ in this kernel mapping to a $k$-basis of $H^0(E, \mathcal{O}_ E(-E)) = H^0(\mathbf{P}^1_ k, \mathcal{O}(1))$. Then $x^ d, x^{d - 1}y, \ldots , y^ d$ are elements of $\mathop{\mathrm{lim}}\nolimits H^0(E_ n, \mathcal{O}_ n(-dE))$ which map to a basis of $H^0(E, \mathcal{O}_ E(-dE)) = H^0(\mathbf{P}^1_ k, \mathcal{O}(d))$. In this way we see that $A$ is separated and complete with respect to the linear topology defined by the kernels

We have $x, y \in I_1$, $I_ d I_{d'} \subset I_{d + d'}$ and $I_ d/I_{d + 1}$ is a free $k$-module on $x^ d, x^{d - 1}y, \ldots , y^ d$. We will show that $I_ d = (x, y)^ d$. Namely, if $z_ e \in I_ e$ with $e \geq d$, then we can write

where $a_{e, j} \in (x, y)^{e - d}$ and $z_{e + 1} \in I_{e + 1}$ by our description of $I_ d/I_{d + 1}$. Thus starting with some $z = z_ d \in I_ d$ we can do this inductively

with some $a_{e, j} \in (x, y)^{e - d}$. Then $a_ j = \sum _{e \geq d} a_{e, j}$ exists (by completeness and the fact that $a_{e, j} \in I_{e - d}$) and we have $z = \sum a_{e, j} x^{d - j} y^ j$. Hence $I_ d = (x, y)^ d$. Thus $A$ is $(x, y)$-adically complete. Then $A$ is Noetherian by Algebra, Lemma 10.97.5. It is clear that the dimension is $2$ by the description of $(x, y)^ d/(x, y)^{d + 1}$ and Algebra, Proposition 10.60.8. Since the maximal ideal is generated by two elements it is regular. $\square$

Lemma 54.16.4. Let $X$ be a Noetherian scheme. Let $E \subset X$ be an exceptional curve of the first kind. If there exists a morphism $f : X \to Y$ such that

$Y$ is Noetherian,

$f$ is proper,

$f$ maps $E$ to a point $y$ of $Y$,

$f$ is quasi-finite at every point not in $E$,

Then there exists a contraction of $E$ and it is the Stein factorization of $f$.

**Proof.**
We apply More on Morphisms, Theorem 37.49.4 to get a Stein factorization $X \to X' \to Y$. Then $X \to X'$ satisfies all the hypotheses of the lemma (some details omitted). Thus after replacing $Y$ by $X'$ we may in addition assume that $f_*\mathcal{O}_ X = \mathcal{O}_ Y$ and that the fibres of $f$ are geometrically connected.

Assume that $f_*\mathcal{O}_ X = \mathcal{O}_ Y$ and that the fibres of $f$ are geometrically connected. Note that $y \in Y$ is a closed point as $f$ is closed and $E$ is closed. The restriction $f^{-1}(Y \setminus \{ y\} ) \to Y \setminus \{ y\} $ of $f$ is a finite morphism (More on Morphisms, Lemma 37.40.1). Hence this restriction is an isomorphism since $f_*\mathcal{O}_ X = \mathcal{O}_ Y$ since finite morphisms are affine. To prove that $\mathcal{O}_{Y, y}$ is regular of dimension $2$ we consider the isomorphism

of Cohomology of Schemes, Lemma 30.20.7. Let $E_ n = nE$ as in Lemma 54.16.3. Observe that

because $E \subset X_ y = X \times _ Y \mathop{\mathrm{Spec}}(\kappa (y))$. On the other hand, since $E = f^{-1}(\{ y\} )$ set theoretically (because the fibres of $f$ are geometrically connected), we see that the scheme theoretic fibre $X_ y$ is scheme theoretically contained in $E_ n$ for some $n > 0$. Namely, apply Cohomology of Schemes, Lemma 30.10.2 to the coherent $\mathcal{O}_ X$-module $\mathcal{F} = \mathcal{O}_{X_ y}$ and the ideal sheaf $\mathcal{I}$ of $E$ and use that $\mathcal{I}^ n$ is the ideal sheaf of $E_ n$. This shows that

Thus the inverse limit displayed above is equal to $\mathop{\mathrm{lim}}\nolimits H^0(E_ n, \mathcal{O}_ n)$ which is a regular two dimensional local ring by Lemma 54.16.3. Hence $\mathcal{O}_{Y, y}$ is a two dimensional regular local ring because its completion is so (More on Algebra, Lemma 15.43.4 and 15.43.1).

We still have to prove that $f : X \to Y$ is the blowup $b : Y' \to Y$ of $Y$ at $y$. We encourage the reader to find her own proof. First, we note that Lemma 54.16.3 also implies that $X_ y = E$ scheme theoretically. Since the ideal sheaf of $E$ is invertible, this shows that $f^{-1}\mathfrak m_ y \cdot \mathcal{O}_ X$ is invertible. Hence we obtain a factorization

of the morphism $f$ by the universal property of blowing up, see Divisors, Lemma 31.32.5. Recall that the exceptional fibre of $E' \subset Y'$ is an exceptional curve of the first kind by Lemma 54.3.1. Let $g : E \to E'$ be the induced morphism. Because for both $E'$ and $E$ the conormal sheaf is generated by (pullbacks of) $a$ and $b$, we see that the canonical map $g^*\mathcal{C}_{E'/Y'} \to \mathcal{C}_{E/X}$ (Morphisms, Lemma 29.31.3) is surjective. Since both are invertible, this map is an isomorphism. Since $\mathcal{C}_{E/X}$ has positive degree, it follows that $g$ cannot be a constant morphism. Hence $g$ has finite fibres. Hence $g$ is a finite morphism (same reference as above). However, since $Y'$ is regular (and hence normal) at all points of $E'$ and since $X \to Y'$ is birational and an isomorphism away from $E'$, we conclude that $X \to Y'$ is an isomorphism by Varieties, Lemma 33.17.3. $\square$

Lemma 54.16.5. Let $b : X \to X'$ be the contraction of an exceptional curve of the first kind $E \subset X$. Then there is a short exact sequence

where the first map is pullback by $b$ and the second map sends $\mathcal{L}$ to the degree of $\mathcal{L}$ on the exceptional curve $E$. The sequence is split by the map $n \mapsto \mathcal{O}_ X(-nE)$.

**Proof.**
Since $E = \mathbf{P}^1_ k$ we see that the Picard group of $E$ is $\mathbf{Z}$, see Divisors, Lemma 31.28.5. Hence we can think of the last map as $\mathcal{L} \mapsto \mathcal{L}|_ E$. The degree of the restriction of $\mathcal{O}_ X(E)$ to $E$ is $-1$ by definition of exceptional curves of the first kind. Combining these remarks we see that it suffices to show that $\mathop{\mathrm{Pic}}\nolimits (X') \to \mathop{\mathrm{Pic}}\nolimits (X)$ is injective with image the invertible sheaves restricting to $\mathcal{O}_ E$ on $E$.

Given an invertible $\mathcal{O}_{X'}$-module $\mathcal{L}'$ we claim the map $\mathcal{L}' \to b_*b^*\mathcal{L}'$ is an isomorphism. This is clear everywhere except possibly at the image point $x \in X'$ of $E$. To check it is an isomorphism on stalks at $x$ we may replace $X'$ by an open neighbourhood at $x$ and assume $\mathcal{L}'$ is $\mathcal{O}_{X'}$. Then we have to show that the map $\mathcal{O}_{X'} \to b_*\mathcal{O}_ X$ is an isomorphism. This follows from Lemma 54.3.4 part (4).

Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module with $\mathcal{L}|_ E = \mathcal{O}_ E$. Then we claim (1) $b_*\mathcal{L}$ is invertible and (2) $b^*b_*\mathcal{L} \to \mathcal{L}$ is an isomorphism. Statements (1) and (2) are clear over $X' \setminus \{ x\} $. Thus it suffices to prove (1) and (2) after base change to $\mathop{\mathrm{Spec}}(\mathcal{O}_{X', x})$. Computing $b_*$ commutes with flat base change (Cohomology of Schemes, Lemma 30.5.2) and similarly for $b^*$ and formation of the adjunction map. But if $X'$ is the spectrum of a regular local ring then $\mathcal{L}$ is trivial by the description of the Picard group in Lemma 54.3.3. Thus the claim is proved.

Combining the claims proved in the previous two paragraphs we see that the map $\mathcal{L} \mapsto b_*\mathcal{L}$ is an inverse to the map

and the lemma is proved. $\square$

Remark 54.16.6. Let $b : X \to X'$ be the contraction of an exceptional curve of the first kind $E \subset X$. From Lemma 54.16.5 we obtain an identification

where $\mathcal{L}$ corresponds to the pair $(\mathcal{L}', n)$ if and only if $\mathcal{L} = (b^*\mathcal{L}')(-nE)$, i.e., $\mathcal{L}(nE) = b^*\mathcal{L}'$. In fact the proof of Lemma 54.16.5 shows that $\mathcal{L}' = b_*\mathcal{L}(nE)$. Of course the assignment $\mathcal{L} \mapsto \mathcal{L}'$ is a group homomorphism.

Lemma 54.16.7. Let $X$ be a Noetherian scheme. Let $E \subset X$ be an exceptional curve of the first kind. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $n$ be the integer such that $\mathcal{L}|_ E$ has degree $n$ viewed as an invertible module on $\mathbf{P}^1$. Then

If $H^1(X, \mathcal{L}) = 0$ and $n \geq 0$, then $H^1(X, \mathcal{L}(iE)) = 0$ for $0 \leq i \leq n + 1$.

If $n \leq 0$, then $H^1(X, \mathcal{L}) \subset H^1(X, \mathcal{L}(E))$.

**Proof.**
Observe that $\mathcal{L}|_ E = \mathcal{O}(n)$ by Divisors, Lemma 31.28.5. Use induction, the long exact cohomology sequence associated to the short exact sequence

and use the fact that $H^1(\mathbf{P}^1, \mathcal{O}(d)) = 0$ for $d \geq -1$ and $H^0(\mathbf{P}^1, \mathcal{O}(d)) = 0$ for $d \leq -1$. Some details omitted. $\square$

Lemma 54.16.8. Let $S = \mathop{\mathrm{Spec}}(R)$ be an affine Noetherian scheme. Let $X \to S$ be a proper morphism. Let $\mathcal{L}$ be an ample invertible sheaf on $X$. Let $E \subset X$ be an exceptional curve of the first kind. Then

there exists a contraction $b : X \to X'$ of $E$,

$X'$ is proper over $S$, and

the invertible $\mathcal{O}_{X'}$-module $\mathcal{L}'$ is ample with $\mathcal{L}'$ as in Remark 54.16.6.

**Proof.**
Let $n$ be the degree of $\mathcal{L}|_ E$ as in Lemma 54.16.7. Observe that $n > 0$ as $\mathcal{L}$ is ample on $E$ (Varieties, Lemma 33.43.14 and Properties, Lemma 28.26.3). After replacing $\mathcal{L}$ by a power we may assume $H^ i(X, \mathcal{L}^{\otimes e}) = 0$ for all $i > 0$ and $e > 0$, see Cohomology of Schemes, Lemma 30.17.1. Finally, after replacing $\mathcal{L}$ by another power we may assume there exist global sections $t_0, \ldots , t_ n$ of $\mathcal{L}$ which define a closed immersion $\psi : X \to \mathbf{P}^ n_ S$, see Morphisms, Lemma 29.39.4.

Set $\mathcal{M} = \mathcal{L}(nE)$. Then $\mathcal{M}|_ E \cong \mathcal{O}_ E$. Since we have the short exact sequence

and since $H^1(X, \mathcal{M}(-E))$ is zero (by Lemma 54.16.7 and the fact that $n > 0$) we can pick a section $s_{n + 1}$ of $\mathcal{M}$ which generates $\mathcal{M}|_ E$. Finally, denote $s_0, \ldots , s_ n$ the sections of $\mathcal{M}$ we get from the sections $t_0, \ldots , t_ n$ of $\mathcal{L}$ chosen above via $\mathcal{L} \subset \mathcal{L}(nE) = \mathcal{M}$. Combined the sections $s_0, \ldots , s_ n, s_{n + 1}$ generate $\mathcal{M}$ in every point of $X$ and therefore define a morphism

over $S$, see Constructions, Lemma 27.13.1.

Below we will check the conditions of Lemma 54.16.4. Once this is done we see that the Stein factorization $X \to X' \to \mathbf{P}^{n + 1}_ S$ of $\varphi $ is the desired contraction which proves (1). Moreover, the morphism $X' \to \mathbf{P}^{n + 1}_ S$ is finite hence $X'$ is proper over $S$ (Morphisms, Lemmas 29.44.11 and 29.41.4). This proves (2). Observe that $X'$ has an ample invertible sheaf. Namely the pullback $\mathcal{M}'$ of $\mathcal{O}_{\mathbf{P}^{n + 1}_ S}(1)$ is ample by Morphisms, Lemma 29.37.7. Observe that $\mathcal{M}'$ pulls back to $\mathcal{M}$ on $X$ (by Constructions, Lemma 27.13.1). Finally, $\mathcal{M} = \mathcal{L}(nE)$. Since in the arguments above we have replaced the original $\mathcal{L}$ by a positive power we conclude that the invertible $\mathcal{O}_{X'}$-module $\mathcal{L}'$ mentioned in (3) of the lemma is ample on $X'$ by Properties, Lemma 28.26.2.

Easy observations: $\mathbf{P}^{n + 1}_ S$ is Noetherian and $\varphi $ is proper. Details omitted.

Next, we observe that any point of $U = X \setminus E$ is mapped to the open subscheme $W$ of $\mathbf{P}^{n + 1}_ S$ where one of the first $n + 1$ homogeneous coordinates is nonzero. On the other hand, any point of $E$ is mapped to a point where the first $n + 1$ homogeneous coordinates are all zero, in particular into the complement of $W$. Moreover, it is clear that there is a factorization

of $\psi |_ U$ where $pr$ is the projection using the first $n + 1$ coordinates and $\psi : X \to \mathbf{P}^ n_ S$ is the embedding chosen above. It follows that $\varphi |_ U : U \to W$ is quasi-finite.

Finally, we consider the map $\varphi |_ E : E \to \mathbf{P}^{n + 1}_ S$. Observe that for any point $x \in E$ the image $\varphi (x)$ has its first $n + 1$ coordinates equal to zero, i.e., the morphism $\varphi |_ E$ factors through the closed subscheme $\mathbf{P}^0_ S \cong S$. The morphism $E \to S = \mathop{\mathrm{Spec}}(R)$ factors as $E \to \mathop{\mathrm{Spec}}(H^0(E, \mathcal{O}_ E)) \to \mathop{\mathrm{Spec}}(R)$ by Schemes, Lemma 26.6.4. Since by assumption $H^0(E, \mathcal{O}_ E)$ is a field we conclude that $E$ maps to a point in $S \subset \mathbf{P}^{n + 1}_ S$ which finishes the proof. $\square$

Lemma 54.16.9. Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a morphism of finite type. Let $E \subset X$ be an exceptional curve of the first kind which is in a fibre of $f$.

If $X$ is projective over $S$, then there exists a contraction $X \to X'$ of $E$ and $X'$ is projective over $S$.

If $X$ is quasi-projective over $S$, then there exists a contraction $X \to X'$ of $E$ and $X'$ is quasi-projective over $S$.

**Proof.**
Both cases follow from Lemma 54.16.8 using standard results on ample invertible modules and (quasi-)projective morphisms.

Proof of (2). Projectivity of $f$ means that $f$ is proper and there exists an $f$-ample invertible module $\mathcal{L}$, see Morphisms, Lemma 29.43.13 and Definition 29.40.1. Let $U \subset S$ be an affine open containing the image of $E$. By Lemma 54.16.8 there exists a contraction $c : f^{-1}(U) \to V'$ of $E$ and an ample invertible module $\mathcal{N}'$ on $V'$ whose pullback to $f^{-1}(U)$ is equal to $\mathcal{L}(nE)|_{f^{-1}(U)}$. Let $v \in V'$ be the closed point such that $c$ is the blowing up of $v$. Then we can glue $V'$ and $X \setminus E$ along $f^{-1}(U) \setminus E = V' \setminus \{ v\} $ to get a scheme $X'$ over $S$. The morphisms $c$ and $\text{id}_{X \setminus E}$ glue to a morphism $b : X \to X'$ which is the contraction of $E$. The inverse image of $U$ in $X'$ is proper over $U$. On the other hand, the restriction of $X' \to S$ to the complement of the image of $v$ in $S$ is isomorphic to the restriction of $X \to S$ to that open. Hence $X' \to S$ is proper (as being proper is local on the base by Morphisms, Lemma 29.41.3). Finally, $\mathcal{N}'$ and $\mathcal{L}|_{X \setminus E}$ restrict to isomorphic invertible modules over $f^{-1}(U) \setminus E = V' \setminus \{ v\} $ and hence glue to an invertible module $\mathcal{L}'$ over $X'$. The restriction of $\mathcal{L}'$ to the inverse image of $U$ in $X'$ is ample because this is true for $\mathcal{N}'$. For affine opens of $S$ avoiding the image of $v$, we see that the same is true because it holds for $\mathcal{L}$. Thus $\mathcal{L}'$ is $(X' \to S)$-relatively ample by Morphisms, Lemma 29.37.4 and (2) is proved.

Proof of (3). We can write $X$ as an open subscheme of a scheme $\overline{X}$ projective over $S$ by Morphisms, Lemma 29.43.12. By (2) there is a contraction $b : \overline{X} \to \overline{X}'$ and $\overline{X}'$ is projective over $S$. Then we let $X' \subset \overline{X}$ be the image of $X \to \overline{X}'$; this is an open as $b$ is an isomorphism away from $E$. Then $X \to X'$ is the desired contraction. Note that $X'$ is quasi-projective over $S$ as it has an $S$-relatively ample invertible module by the construction in the proof of part (2). $\square$

Lemma 54.16.10. Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a separated morphism of finite type with $X$ regular of dimension $2$. Then $X$ is quasi-projective over $S$.

**Proof.**
By Chow's lemma (Cohomology of Schemes, Lemma 30.18.1) there exists a proper morphism $\pi : X' \to X$ which is an isomorphism over a dense open $U \subset X$ such that $X' \to S$ is H-quasi-projective. By Lemma 54.4.3 there exists a sequence of blowups in closed points

and an $S$-morphism $X_ n \to X'$ extending the rational map $U \to X'$. Observe that $X_ n \to X$ is projective by Divisors, Lemma 31.32.13 and Morphisms, Lemma 29.43.14. This implies that $X_ n \to X'$ is projective by Morphisms, Lemma 29.43.15. Hence $X_ n \to S$ is quasi-projective by Morphisms, Lemma 29.40.3 (and the fact that a projective morphism is quasi-projective, see Morphisms, Lemma 29.43.10). By Lemma 54.16.9 (and uniqueness of contractions Lemma 54.16.2) we conclude that $X_{n - 1}, \ldots , X_0 = X$ are quasi-projective over $S$ as desired. $\square$

Lemma 54.16.11. Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a proper morphism with $X$ regular of dimension $2$. Then $X$ is projective over $S$.

**Proof.**
This follows from Lemma 54.16.10 and Morphisms, Lemma 29.43.13.
$\square$

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