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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$\begingroup$ Closely related: mathoverflow.net/questions/152/… $\endgroup$– Qiaochu YuanCommented Jun 17, 2010 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.
answered Jun 17, 2010 at 10:23
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$\begingroup$ 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. $\endgroup$ Commented Jun 17, 2010 at 10:31
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$\begingroup$ 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. $\endgroup$ Commented Jun 17, 2010 at 13:00
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1$\begingroup$ 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. $\endgroup$ Commented Jun 17, 2010 at 14:32
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2$\begingroup$ @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 $h^1(\mathcal{O})$! $\endgroup$– BoyarskyCommented Jun 18, 2010 at 1:45