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For me a Del Pezzo surface $X$ over an algebraically closed field of characteristic $p$ is an algebraic surface where the anticanonical bundle $\omega^{-1}_X$ or $-K_X$ is ample. (I prefer the second notation, although it is not very correct),

In characteristic 0, as far as I know, there is a classification. $X$ has to be isomporphic either to $\mathbb{P}^1\times \mathbb{P}^1$ or a blow-up of $\mathbb{P}^2$ at $9-K_X^2\geq 0$ points in general position, i.e. not 3 points in the same line and not 6 points in the same conic of $\mathbb{P}^2$.

I would like to know if there is such a classification in characteristic $p\geq 2$. As far as I know the classification is definitely the same for $p>3$, and probably even for $p=3$ and there are 'extra' surfaces for $p=2$.

The perfect answer would confirm whether these assertions are true, describe the extra cases (in particular in terms of which curves can live in them or minimality) and/or give a reference.

But anything is better than nothing, so even if you know a bit of this I would like to know.

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The case of $X={\mathbb P}^1\times {\mathbb P}^1$ is missing in your list. –  rita Sep 23 '11 at 14:05
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Singular del Pezzo surfaces are also of interest; see "Non-normal del Pezzo surfaces" by Reid (MR1311389). Extra things do happen in char. p. –  inkspot Sep 23 '11 at 16:36
    
Fixed the typo, thanks rita –  Jesus Martinez Garcia Sep 23 '11 at 19:01

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up vote 8 down vote accepted

The classical classification of (smooth) Del Pezzo surfaces as blow-ups relies on the Kodaira vanishing theorem in characteristic zero, but is actually true over any algebraically closed field. See for example Kollar's Rational curves on algebraic varieties book, section III.3. (This paper by Xie on the Kawamata vanishing theorem on rational surfaces in characteristic $p$ might also be useful.)

There is also an interesting classification of Del Pezzo surfaces as complete intersections in weighted projective spaces, which holds over any base field (not neccessarily algebraically closed). Again, the reference is Kollar's book, chapter III.3.

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