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Is there a classification of surfaces(smooth and projective) over arbitrary field? Whether using the approach of Enriques or not. thanks

P.S. By arbitrary I mean the field may not be algebraic closed, even not perfect, since as far as I know variety over perfect field is much like one over closed field. So Is there a treatment on the case of non-perfect case. Thanks

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What is a smooth surface over an arbitrary field? –  Matthias Ludewig Feb 24 '12 at 14:39
    
I mean a smooth surface X/k if the structure morphism X to k is smooth –  stjc Feb 26 '12 at 10:12

3 Answers 3

Try Wikipedia http://en.wikipedia.org/wiki/Enriques%E2%80%93Kodaira_classification: they say that the classification was begun by Mumford, and completed by Mumford and Bombieri, and they give references. They say ``it is similar to the characteristic projective 0 case, except there are a few extra types of surface in characteristics 2 and 3.''

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You probably want to work over an algebraically closed field, at least initially. For surfaces in positive characteristic, have a look at these very nice notes by Christian Liedtke: Algebraic Surfaces in Positive Characteristic.

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thanks a lot, I didn't make the question clear, but I want to know surface over non algebraic closed field in particular:) –  stjc Feb 26 '12 at 10:37

One of the main subtleties in trying to classify surfaces over non-algebraically closed fields is that there are minimal surfaces which become non-minimal over the algebraic closure.

As an example I will focus on the case that I know best, that of (geometrically) rational surfaces. Over an algebraically closed field, it is well-known that the only such minimal surfaces are $\mathbb{P}^2$ and the rational ruled surfaces $\mathbb{F}_n$ for $n \geq 0$.

If the field is not algebraically closed, then things are a lot more complicated. It is a theorem of Iskovskikh that a minimal rational surface over a perfect field is one of the following types:

  • $\mathbb{P}^2$.
  • A smooth quadric $X \subset \mathbb{P}^3$ with $\mathrm{Pic}(X) = \mathbb{Z}$.
  • A Del Pezzo surface $X$ with $\mathrm{Pic}(X) = \mathbb{Z}K_X$, here $K_X$ denotes the canonical divisor.
  • A conic bundle $f : X \to C$ over a rational curve $C$, with $\mathrm{Pic}(X) = \mathbb{Z} \oplus \mathbb{Z}$.

In particular conic bundles form a very large family and can have arbitrarily many (geometrically) degenerate fibres.

If you want to learn more about this result, I heartily recommend the notes "Rational surfaces over nonclosed fields" by Brendan Hassett, which can be found on his webpage.

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