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 10.60.9. 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 37.53.4 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 37.44.1). 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 33.44.14 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 $\text{pr}$ 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 (1). 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 (1) is proved.

Proof of (2). We can write $X$ as an open subscheme of a scheme $\overline{X}$ projective over $S$ by Morphisms, Lemma 29.43.12. By (1) 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 (1). $\square$

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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)