Let $L$ be an ample line bundle in $\mathbb{P}^n$, with at least $n$ global sections. Choose two sets of $n$ linearly independent global sections of $L$, say $S\_1:=\{D\_1,...,D\_n\}$ and $S\_2:=\{E\_1,....,E\_n\}$, where $D\_i$ and $E\_i$ are effective divisors corresponding to global sections on $L$. Does there exist a Cremona / birational transformation $f: \mathbb{P}^n \dashrightarrow \mathbb{P}^n$ sending $D\_i$ to $E\_i$ (birationally) for each $i$?



Cremona transformations and divisors
In general, I think that the answer is negative. In fact, any birational transformation must preserve the geometric genus of divisors, but this is not fixed in the linear system $|L|$.


For instance, take $n=2$ and $L=\mathcal{O}(3)$. Let $D\_1, \, D\_2$ be smooth plane cubics and let $E\_1$ be a nodal cubic. There is no birational map between $D\_i$ $(i=1,\, 2)$ and $E\_1$, because $D\_i$ has geometric genus $1$, whereas $E\_1$ is rational.


