Let $S$ be a smooth, projective surface over an algebraically closed field $k$ of any characteristic.
I'm trying to prove/disprove the following:
There cannot be a sequence of curves $\{E_n\}_{n\in\Bbb{N}}$ on $S$ such that:
(i) each $E_i$ is isomorphic to $\Bbb{P}^1$ with $E_i^2=-1$.
(ii) $E_i\cdot E_j=0$ whenever $i\neq j$.
I know a proof for this when $k=\Bbb{C}$. Using Castelnuovo's conctractibility criterion, we can successively contract $E_1,E_2,...$ through blowdowns $S=:S_0\to S_1\to S_2\to...$
After we contract $E_n$, we still have $(-1)$-curves $E_{n+1},E_{n+2},...$ on $S_n$, so we never obtain a minimal model, which is absurd.
Is it possible to use a similar argument for arbitrary algebraically closed fields, including positive characteristic?
(I don't know if this will make a difference, but in the original context of this, $S$ is rational)
Thanks you!
I suppose my comment can function as an answer, so let me repost it here: Castelnuovo's criteria is fully valid in characteristic $p>0$ - the proof in Hartshorne (V.5.7) is completely characteristic independent. Since blowing down a $-1$ curve drops (etale) $b_2$ by $1$ every time you do it, you indeed get a minimal model.