# Hartshorne Exercise V.5.5 Every geometrically ruled surface over a fixed curve $C$ is birationally equivalent

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...