# Is there a classification of surface(smooth and projective) over arbitrary field?

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

• 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

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.

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.

• 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

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.''