Cremona transformations and divisors

Question

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 $i$?

Answer 1

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.