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genus (Definition)

``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.




"genus" is owned by mathcam. [ full author list (3) ]
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Hurwitz genus formula (Theorem) by alozano
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Cross-references: Riemann surface, curve, complex, regular, vector space, dimension, field, projective curve, smooth, algebraic geometry, Euler characteristic, torus, additive, connected sum, sphere, surface, orientable, topology, definitions, compatible, number
There are 21 references to this entry.

This is version 6 of genus, born on 2001-12-21, modified 2002-12-04.
Object id is 1116, canonical name is Genus.
Accessed 8648 times total.

Classification:
AMS MSC14H99 (Algebraic geometry :: Curves :: Miscellaneous)

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standard definition of the genus by jocaps on 2009-03-19 16:49:16
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.
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Notification before going to Orphanage by alozano on 2004-02-25 13:12:32

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
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change of name? by Dr_Absentius on 2002-08-20 11:50:37
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 ...).
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