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Let $K$ be an algebraic function field of one variable. Then we can define its genus. On the other hand, it can also be seen as a scheme, so we can define the arithmetic and geometric genus. Could anyone please tell me the relation between these definitions?

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Closely related: mathoverflow.net/questions/152/… –  Qiaochu Yuan Jun 17 '10 at 21:53
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The definitions coincide, with some caveats: basically for a curve, there is a single notion of genus, which applies equally to smooth curves over algebraically closed fields, and to their function fields; and also over the complex numbers to the associated Riemann surface as two-dimensional manifold. See http://en.wikipedia.org/wiki/Genus_%28mathematics%29 . On the other hand care is needed for curves that are allowed to be singular, or fields that are not algebraically closed, what definition is in use.

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I feel I should mention the phenomenon of "genus change in purely inseparable extension", given that this is something surprising in characteristic p; but I don't know whether there is a definitive treatment out there in what is a rather large literature by now. –  Charles Matthews Jun 17 '10 at 10:31
    
Thx Charles. I am not sure what you mean by "genus change". As I know, for function fields, the genus of an algebraic extension field can be computed by Hurwitz formula. –  Yujia Qiu Jun 17 '10 at 13:00
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This is about inseparable coverings; there is a classic paper of John Tate I've never read (fortunately I've now found it is online) which refers to Emil Artin's concept of a "conservative function field", where the genus doesn't change under extension of constant field. Tate's paper seems to use the Cartier operator (before Cartier); I imagine this isse is now well understood. Could be another question, though. –  Charles Matthews Jun 17 '10 at 14:32
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@Charles: This is in Artin's book "Algebraic numbers and algebraic functions"; his index points to the def'n and the discussion of behavior under extension of the constant field (at end of Ch. 15 he gives the Tate paper reference). @Yujia: The issue is that if $X$ is a regular proper geom. integral curve over a field $k$ (this is the geometric object intrinsic to a trdeg 1 function field over $k$) and if $k'/k$ is an extension then the base change $X'=X_ {k'}$ is a proper integral curve over $k$ which may not be regular. So its normalization may have different $h^1(\mathcal{O})$! –  Boyarsky Jun 18 '10 at 1:45
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