I’m reading the proof of the Castelnuovo’s contractibility criterion in Beauville’s book(Theorem II.17), and I guess I could understand all its affirmations. But I still has one question.

For those who don’t remember, Castelnuovo’s contractibility criterion states that “a (smooth, projective, complex) surface $S$ with a curve $E$ such that $E simeq mathbb{P}^1$ and $E^2=-1$ is the blow-up of another surface, and $E$ is the exceptional curve of this blow-up”.

The proof of this is to construct the blow-down map $phi:S rightarrow S’$, and then prove that $S’$ is also smooth.

The map is constructed by linear systems. Beauville’s choose a lot of thins:

- a hyperplane section $H$
- an integer $k$
- a section $sin mathcal{O}_S(E)$ such that $(s)_0 = E$
- a basis $s_0,ldots,s_n$ of $H^0(S, mathcal{O}_S(H))$
- a basis $a_{i,0},ldots,a_{i,k-i}$ of $H^0(S, mathcal{O}_S(H+iE))$ for $1le i le k$

With the chooses done correctly, he says that

$${s^ks_0,ldots,s^ks_n,s^{k-1}a_{1,0},ldots,s^{k-1}a_{1,k-1},ldots,sa_{k-1,1},a_{k,0} } $$

is a basis of $H^0(S, mathcal{O}_S(H’) )$, where $H’ = H + kE$. Then, he constructs a rational map $phi: S dashrightarrow mathbb{P}^N$, and want to prove that this map is a blow-down to a smooth surface. That is:

- $phi$ is a embedding at $S-E$;
- $phi(E)$ is a point;
- $phi(S)$ is a smooth surface.

And it’s done at the proof. **My doubt arises here**: I understand that this map works for the proof, but I think that it can be simplified. I want to consider $psi: S dashrightarrow mathbb{P}^N$ given by the subspace of $|H’|$ with basis ${s^k s_0,ldots,s^ks_0, s a_{k-1,0}, s a_{k-1,1} , a_{k,0} }$. I guess this is enough to prove the three affirmations:

- $psi$ is an embedding at $S-E$, because $s$ is invertible out of $E$, and as $(s_0: cdots :s_n)$ defines an embedding, then $psi$ also defines at $S-E$
- $psi(E)$ is a point, because $(s)_0 = E$ and then $phi(E) = (0:cdots:0:1)$ ( $a_{k,0}$ is constructed such that $(a_{k,0})_0 cap E = emptyset$)
- $psi(S)$ is smooth; first, we just need to prove that $psi(S)$ is smooth at $(0:cdots:0:1)$, because $psi(S-E)$ is already smooth. Beauville do this putting coordinates and do a lot of considerations, but uses just the three last coordinates to do this.

I just want to know if this is right. If it is, I don’t think this is important, because I always like to use the full linear system anyway. But if it’s wrong, I would like to know, because I’m probably missing something.