## V.5: Birational Transformations 

### V.5.1.  #to-work

Let $f$ be a rational function on the surface $X$. Show that it is possible to "resolve the singularities of $f^{\prime \prime}$ in the following sense: there is a birational morphism $g$ : $X^{\prime} \rightarrow X$ so that $f$ induces a morphism of $X^{\prime}$ to $\mathbf{P}^1$.[^hit.v.5.a]

[^hit.v.5.a]: 
Hints: Write the divisor of $f$ as $(f)=\sum n_i C_i$. Then apply embedded resolution (3.9) to the curve $Y=\bigcup C_i$. Then blow up further as necessary whenever a curve of zeros meets a curve of poles until the zeros and poles of $f$ are disjoint. 


### V.5.2.  #to-work

Let $Y \cong \mathbf{P}^1$ be a curve in a surface $X$, with $Y^2<0$. Show that $Y$ is contractible (5.7.2) to a point on a projective variety $X_0$ (in general singular).

### V.5.3.  #to-work

If $\pi: \tilde{X} \rightarrow X$ is a monoidal transformation with center $P$, show that $H^1\left(\tilde{X}, \Omega_{\tilde{X}}\right) \cong$ $H^1\left(X, \Omega_X\right) \oplus k$. This gives another proof of (5.8).[^hit.v.5.3]

[^hit.v.5.3]: 
Hints: Use the projection formula (III, Ex. 8.3) and (III, Ex. 8.1) to show that $H^i\left(X, \Omega_X\right) \cong H^i\left(\tilde{X}, \pi^* \Omega_X\right)$ for each i. Next use the exact sequence
\[
0 \rightarrow \pi^* \Omega_X \rightarrow \Omega_{\tilde{X}} \rightarrow \Omega_{\tilde{X} / X} \rightarrow 0
\]
and a local calculation with coordinates to show that there is a natural isomorphism $\Omega_{\tilde{X} / X} \cong \Omega_E$, where $E$ is the exceptional curve. Now use the cohomology sequence of the above sequence (you will need every term) and Serre duality to get the result.


### V.5.4.  #to-work

Let $f: X \rightarrow X^{\prime}$ be a birational morphism of nonsingular surfaces.

a. If $Y \subseteq X$ is an irreducible curve such that $f(Y)$ is a point, then $Y \cong \mathbf{P}^1$ and $Y^2<0$

b. Let $P^{\prime} \in X^{\prime}$ be a fundamental point of $f^{-1}$, and let $Y_1, \ldots, Y_r$ be the irreducible components of $f^{-1}\left(P^{\prime}\right)$. Show that the matrix $\left\|Y_i . Y_j\right\|$ is negative definite.

### V.5.5.  #to-work

Let $C$ be a curve, and let $\pi: X \rightarrow C$ and $\pi^{\prime}: X^{\prime} \rightarrow C$ be two geometrically ruled surfaces over $C$. Show that there is a finite sequence of elementary transformations (5.7.1) which transform $X$ into $X^{\prime}$.[^hint.v.5.5]

[^hint.v.5.5]: 
Hints: First show if $D \subseteq X$ is a section of $\pi$ containing a point $P$, and if $\tilde{D}$ is the strict transform of $D$ by $\mathrm{elm}_P$, then $\widetilde{D}^2=D^2-1$ (Fig. 23). 
\
Next show that $X$ can be transformed into a geometrically ruled surface $X^{\prime \prime}$ with invariant $e \gg 0$. Then use (2.12), and study how the ruled surface $\mathbf{P}(\mathcal{E})$ with $\mathcal{E}$ decomposable behaves under $\operatorname{elm}_P$.


### V.5.6.  #to-work

Let $X$ be a surface with function field $K$. Show that every valuation $\operatorname{ring} R$ of $K / k$ is one of the three kinds described in (II, Ex. 4.12).[^hint.v.5.6]

[^hint.v.5.6]: 
Hint: In case (3), let $f \in R$. Use (Ex. 5.1) to show that for all $i \gg 0, f \in \mathcal{O}_{X_i}$, so in fact $f \in R_0$.


### V.5.7.  #to-work

Let $Y$ be an irreducible curve on a surface $X$, and suppose there is a morphism $f: X \rightarrow X_0$ to a projective variety $X_0$ of dimension 2 , such that $f(Y)$ is a point $P$ and $f^{-1}(P)=Y$. Then show that $Y^2<0$.[^hint.v.5.7]

[^hint.v.5.7]: 
Hint: Let $|H|$ be a very ample (Cartier) divisor class on $X_0$, let $H_0 \in|H|$ be a divisor containing $P$, and let $H_1 \in|H|$ be a divisor not containing $P$. Then consider $f^* H_0, f^* H_1$ and $\tilde{H}_0=f^*\left(H_0-P\right)^{-}$.


### V.5.8. A surface singularity.  #to-work

Let $k$ be an algebraically closed field, and let $X$ be the surface in $\mathbf{A}_k^3$ defined by the equation $x^2+y^3+z^5=0$. It has an isolated singularity at the origin $P=(0,0,0)$.


a. Show that the affine ring $A=k[x, y, z] /\left(x^2+y^3+z^5\right)$ of $X$ is a unique factorization domain, as follows. Let $t=z^{-1} ; u=t^3 x$, and $v=t^2 y$. Show that $z$ is irreducible in $A ; t \in k[u, v]$, and $A\left[z^{-1}\right]=k\left[u, v, t^{-1}\right]$. Conclude that $A$ is a UFD.

b. Show that the singularity at $P$ can be resolved by eight successive blowings-up. If $\tilde{X}$ is the resulting nonsingular surface, then the inverse image of $P$ is a union of eight projective lines, which intersect each other according to the Dynkin $\operatorname{diagram} \mathbf{E}_8$ :
![](Hartshorne_Problems/5_Hartshorne/figures/2022-10-22_21-57-34.png)