Is there a classification of surface(smooth and projective) over arbitrary field? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T07:55:53Z http://mathoverflow.net/feeds/question/89395 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89395/is-there-a-classification-of-surfacesmooth-and-projective-over-arbitrary-field Is there a classification of surface(smooth and projective) over arbitrary field? stjc 2012-02-24T13:25:42Z 2012-02-26T15:15:58Z <p>Is there a classification of surfaces(smooth and projective) over arbitrary field? Whether using the approach of Enriques or not. thanks</p> <p>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</p> http://mathoverflow.net/questions/89395/is-there-a-classification-of-surfacesmooth-and-projective-over-arbitrary-field/89409#89409 Answer by Ben McKay for Is there a classification of surface(smooth and projective) over arbitrary field? Ben McKay 2012-02-24T15:21:35Z 2012-02-24T15:21:35Z <p>Try Wikipedia <a href="http://en.wikipedia.org/wiki/Enriques%E2%80%93Kodaira_classification" rel="nofollow">http://en.wikipedia.org/wiki/Enriques%E2%80%93Kodaira_classification</a>: 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.''</p> http://mathoverflow.net/questions/89395/is-there-a-classification-of-surfacesmooth-and-projective-over-arbitrary-field/89412#89412 Answer by Martin Bright for Is there a classification of surface(smooth and projective) over arbitrary field? Martin Bright 2012-02-24T15:34:13Z 2012-02-24T15:34:13Z <p>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: <a href="http://arxiv.org/abs/0912.4291" rel="nofollow">Algebraic Surfaces in Positive Characteristic</a>.</p> http://mathoverflow.net/questions/89395/is-there-a-classification-of-surfacesmooth-and-projective-over-arbitrary-field/89588#89588 Answer by Daniel Loughran for Is there a classification of surface(smooth and projective) over arbitrary field? Daniel Loughran 2012-02-26T15:15:58Z 2012-02-26T15:15:58Z <p>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.</p> <p>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 $n \geq 0$.</p> <p>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:</p> <ul> <li>$\mathbb{P}^2$.</li> <li>A smooth quadric $X \subset \mathbb{P}^3$ with $\mathrm{Pic}(X) = \mathbb{Z}$.</li> <li>A Del Pezzo surface $X$ with $\mathrm{Pic}(X) = \mathbb{Z}K_X$, here $K_X$ denotes the canonical divisor.</li> <li>A conic bundle $f : X \to C$ over a rational curve $C$, with $\mathrm{Pic}(X) = \mathbb{Z} \oplus \mathbb{Z}$.</li> </ul> <p>In particular conic bundles form a very large family and can have arbitrarily many (geometrically) degenerate fibres.</p> <p>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. </p>