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# genus

“Genus” has number of distinct but compatible definitions.

In topology, if $S$ is an orientable surface, its genus $g(S)$ is the number of “handles” it has. More precisely, from the classification of surfaces, we know that any orientable surface is a sphere, or the connected sum of $n$ tori. We say the sphere has genus 0, and that the connected sum of $n$ tori has genus $n$ (alternatively, genus is additive with respect to connected sum, and the genus of a torus is 1). Also, $g(S)=1-\chi(S)/2$ where $\chi(S)$ is the Euler characteristic of $S$.

In algebraic geometry, the genus of a smooth projective curve $X$ over a field $k$ is the dimension over $k$ of the vector space $\Omega^{1}(X)$ of global regular differentials on $X$. Recall that a smooth complex curve is also a Riemann surface, and hence topologically a surface. In this case, the two definitions of genus coincide.

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## Comments

## change of name?

maybe you should change the name of this definition to "genus of algebraic curve" since genus is more generally an invariant of (real) 2D manifolds. I have submitted a definition of "genus of topological surface" for example.

Also does your definition work over all fields?

If not maybe you should say so (like k is C or R

or ...).

## Notification before going to Orphanage

Hi,

When is an entry taken to adoptable-orphanage, when it has an outstanding correction? I mean, how long does it take until it is taken to orphanage?

I was just wondering if a message could be sent to the author, one day before this happens, just in case the author really wants to keep the entry, and also to notify that this is going to happen sometime soon.

Alvaro

## standard definition of the genus

Your definition of the genus in algebraic geometry. I'd like to see a reference associated to it. I am not sure if this is the standard definition of genus, this may be the original definition of genus in mathematics history but probably not one seen in most mathematical reference. Could you please cite a reference.

## Re: standard definition of the genus

This is the standard definition of the genus of a smooth projective curve and is sometimes called arithmetic genus, although there is another definition of arithmetic genus and these two don't necessarily coincide for higher dimensional varieties. A reference is Hartshorne's Algebraic Geometry.

## Re: standard definition of the genus

Hi jocaps,

Thanks by you explanation about the concept of ``rank'' in an elliptic curve involving a group structure. I didn't know that.

As far as the definition of elliptic curve, the first time I read that one was here in PM. Please look at:

http://planetmath.org/?op=getobj&from=objects&name=EllipticCurve

Surely you know who is Djao.

perucho