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| ``Genus'' has number of distinct but compatible definitions. |
The {\em genus} of a smooth projective curve $X$ over a field $k$ is the |
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| In topology, if $S$ is an orientable surface, its genus $g(S)$ is the number of ``handles'' it has. |
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| More precisely, from the classification of surfaces, we know that any orientable |
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| surface is a sphere, or the connected sum of $n$ tori. We say the sphere |
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| has genus 0, and that the connected sum of $n$ tori has genus $n$ |
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| (alternatively, genus is additive with respect to connected sum, and the genus of a torus is 1). |
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| Also, $g(S)=1-\chi(S)/2$ where $\chi(S)$ is the Euler characteristic of $S$. |
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| In algebraic geometry, the {genus} of a smooth projective curve $X$ over a field $k$ is the |
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| dimension over $k$ of the vector space $\Omega^1(X)$ of global regular |
dimension over $k$ of the vector space $\Omega^1(X)$ of global regular |
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differentials on $X$. Recall that a smooth complex curve is also a Riemann surface,
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differentials on $X$. |
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and hence topologically a surface. In this case, the two definitions of genus coincide.
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The genus is a birational invariant.
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