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 $X^{\prime}$. [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 elm ${ }_P$, then $\tilde{D}^2=D^2-1$ (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 $\mathbf{P}(\mathscr{E})$ with $\mathscr{E}$ decomposable behaves under $\operatorname{elm}_P$.]

I am currently working on the exercise from Hartshorne above. I can prove the claim that $\widetilde{D}^2=D^2-1$ by using Figure 23 and noting that the computation of this self-intersection can be done on $\widetilde{X}$ in Figure 23 since this is a blow-up. So of course, pulling-back $\widetilde{D}$ is the same thing as the strict transform plus the exceptional divisor.

For the second part regarding the invariant $e\gg 0$, I can use Hartshorne's Proposition V.2.9 to modify a section of $\pi:X\rightarrow C$ so that I can turn $-e$ into an $-e-1$ and so my new, say $e'$, is strictly larger than $e$ from $X\rightarrow C$. So, I can get the invariant $e$ to be sufficienrtly large.

Theorem V.2.12 just says that if $e>2g-2$, then $\mathscr{E}$ is decomposable. So I can reduce to the case where $\pi:X\rightarrow C$ and $\pi':X'\rightarrow C$ are two geometrically ruled surfaces which are projective bundles of some decomposable rank $2$ vector bundles.

How does one describe the behavior of $\mathbb{P}(\mathscr{E})$ under $\operatorname{elm}_P$? I am having trouble doing this particular computation. I want to say that this is a twisting of some factor of $\mathscr{E}$, due to my computations with rational surfaces, but I have been unsuccessful...



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