Let $C$ be a smooth projective curve and $X=\mathbb{P}\_C(E)$ be a ruled surface over $C$.
Let $x\_1,\ x\_2\in X$ be closed points and define $X\_1,\ X\_2$ to be elementary transforms of $X$ at $x\_1,\ x\_2$, respectively.
Then when $X\_1\cong X\_2$ holds? Even if $x\_1,\ x\_2$ are in the same fiber $\pi^{-1}(p)$ where $\pi: X\to C$ and $p\in C$, $X\_1$ and $X\_2$ might be different, because there might be a minimal section passing $x\_1$ but no one passing $x\_2$ for instance. Then the self-intersection numbers of a minimal section for $X\_1$ and $X\_2$ become different. It seems to me very hard to classify the transforms. Is there any result about this question?



When are two elementary transforms isomorphic?
Let $c\_1,c\_2 \in C$ be a pair of points.
The existence of an isomorphism $X\_1 \cong X\_2$ (for some choices of points $x\_1$, $x\_2$ in $X$ over $c\_1$ and $c\_2$) over $C$ is equivalent to the existence of a line bundle $L$ on $C$ of degree 1 such that
$$
L^2 \cong O\_C(c\_1 + c\_2)
$$
and the existence of a morphism $E \otimes L^{-1} \to E$ such that
$$
0 \to E \otimes L^{-1} \to E \to O\_{c\_1} \oplus O\_{c\_2} \to 0.
$$
Indeed, the elementary transformation $X'$ of $X$ at some point $x \in X$ over $c \in C$ comes with a distinguished point $x'$ over $c$ such that the elementary transformation of $X'$ at $x'$ is isomorphic to $X$. Thus, if $X\_1 \cong X\_2$ then the elementary transformation of $X\_1$ at the point $x'\_2$ is isomorphic to $X$, and this is isomorphic to the projectivization of the kernel of an surjective appropriate morphism $E \to O\_{c\_1} \oplus O\_{c\_2}$. The isomorphism $L^2 \cong O\_C(c\_1 + c\_2)$ is obtained by taking the determinant of the above exact sequence.


As an example of a nontrivial situation of that type, let me assume that $L$ is a line bundle such that $L^2 \cong O\_C(c\_1 + c\_2)$ (note that it is determined by the points $c\_1$ and $c\_2$ up to 2-torsion in the Jacobian of $C$) and take
$$
E = O\_C \oplus L^{-1},
$$
so that $E \otimes L^{-1} \cong L^{-1} \oplus L^{-2}$ and consider the morphism
$$
L^{-1} \oplus L^{-2} \to O\_C \oplus L^{-1}
$$
defined by the matrix
$$
\begin{pmatrix} 0 & s \\ 1 & 0 \end{pmatrix},
$$
where $s$ is the section of $L^2$ with zeroes at $c\_1$ and $c\_2$.


In general, for a given vector bundle $E$ the existence of such isomorphisms depends very much on the properties of $E$. For instance, if $\deg(E)$ is even and $E$ is stable, no such nontrivial isomorphisms exist.


