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Difference between revisions of "Genus of a curve"

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A numerical invariant of a one-dimensional [[Algebraic variety|algebraic variety]] defined over a field  $  k $.  
 
A numerical invariant of a one-dimensional [[Algebraic variety|algebraic variety]] defined over a field  $  k $.  
 
The genus of a smooth complete [[Algebraic curve|algebraic curve]]  $  X $
 
The genus of a smooth complete [[Algebraic curve|algebraic curve]]  $  X $
is equal to the dimension of the space of regular differential  $  1 $-
+
is equal to the dimension of the space of regular differential  $  1 $-forms on  $  X $(
forms on  $  X $(
 
 
cf. [[Differential form|Differential form]]). The genus of an algebraic curve  $  X $
 
cf. [[Differential form|Differential form]]). The genus of an algebraic curve  $  X $
 
is equal, by definition, to the genus of the complete algebraic curve birationally isomorphic to  $  X $.  
 
is equal, by definition, to the genus of the complete algebraic curve birationally isomorphic to  $  X $.  
Line 25: Line 24:
 
The algebraic curves of genus  $  g > 1 $
 
The algebraic curves of genus  $  g > 1 $
 
fall into two classes: hyper-elliptic curves and non-hyper-elliptic curves. For non-hyper-elliptic curves  $  X $
 
fall into two classes: hyper-elliptic curves and non-hyper-elliptic curves. For non-hyper-elliptic curves  $  X $
the rational mapping  $  \phi _ {| K _ {X}  | } :  X \rightarrow P  ^ {g-} 1 $
+
the rational mapping  $  \phi _ {| K _ {X}  | } :  X \rightarrow P  ^ {g-1} $
 
defined by the canonical class  $  K _ {X} $
 
defined by the canonical class  $  K _ {X} $
 
of the complete smooth curve is an isomorphic imbedding. For a [[Hyper-elliptic curve|hyper-elliptic curve]]  $  X $
 
of the complete smooth curve is an isomorphic imbedding. For a [[Hyper-elliptic curve|hyper-elliptic curve]]  $  X $

Revision as of 03:40, 29 December 2021


A numerical invariant of a one-dimensional algebraic variety defined over a field $ k $. The genus of a smooth complete algebraic curve $ X $ is equal to the dimension of the space of regular differential $ 1 $-forms on $ X $( cf. Differential form). The genus of an algebraic curve $ X $ is equal, by definition, to the genus of the complete algebraic curve birationally isomorphic to $ X $. For any integer $ g > 0 $ there exists an algebraic curve of genus $ g $. An algebraic curve of genus $ g = 0 $ over an algebraically closed field is a rational curve, i.e. it is birationally isomorphic to the projective line $ P ^ {1} $. Curves of genus $ g = 1 $( elliptic curves, cf. Elliptic curve) are birationally isomorphic to smooth cubic curves in $ P ^ {2} $. The algebraic curves of genus $ g > 1 $ fall into two classes: hyper-elliptic curves and non-hyper-elliptic curves. For non-hyper-elliptic curves $ X $ the rational mapping $ \phi _ {| K _ {X} | } : X \rightarrow P ^ {g-1} $ defined by the canonical class $ K _ {X} $ of the complete smooth curve is an isomorphic imbedding. For a hyper-elliptic curve $ X $ the mapping $ \phi _ {| K _ {X} | } : X \rightarrow P ^ {g - 1 } $ is a two-sheeted covering of a rational curve, $ \phi _ {| K _ {X} | } ( X) $, ramified at $ 2 g + 2 $ points.

If $ X $ is a projective plane curve of degree $ m $, then

$$ g = \frac{( m - 1 ) ( m - 2 ) }{2} - d , $$

where $ d $ is a non-negative integer measuring the deviation from smoothness of $ X $. If $ X $ has only ordinary double points, then $ d $ is equal to the number of singular points of $ X $. For a curve $ X $ in space the following estimate is valid:

$$ g \leq \ \left \{ \begin{array}{ll} \frac{( m - 2 ) ^ {2} }{4} &\textrm{ if } m \textrm{ is even } , \\ \frac{( m - 1 ) ( m - 3 ) }{4} &\textrm{ if } m \textrm{ is odd } , \\ \end{array} \right .$$

where $ m $ is the degree of $ X $ in $ P ^ {3} $.

If $ K= \mathbf C $ is the field of complex numbers, then an algebraic curve $ X $ is the same as a Riemann surface. In this case the smooth complex curve $ X $ of genus $ g $ is homeomorphic to the sphere with $ g $ handles.

References

[1] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[2] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 MR0463157 Zbl 0367.14001

Comments

References

[a1] G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 MR0092855 Zbl 0078.06602
[a2] P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
How to Cite This Entry:
Genus of a curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Genus_of_a_curve&oldid=51791
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article