# algebraic geometry – Contractibility criterion for \$text{char}(k)>0\$?

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!

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