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 ${E_n}_{ninBbb{N}}$ on $S$ such that:

(i) each $E_i$ is isomorphic to $Bbb{P}^1$ with $E_i^2=-1$.

(ii) $E_icdot E_j=0$ whenever $ineq j$.

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 $S=:S_0to S_1to S_2to…$ 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!