Namespaces
Variants
Actions

Difference between revisions of "Genus of a curve"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
m (tex encoded by computer)
Line 1: Line 1:
A numerical invariant of a one-dimensional [[Algebraic variety|algebraic variety]] defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g0439901.png" />. The genus of a smooth complete [[Algebraic curve|algebraic curve]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g0439902.png" /> is equal to the dimension of the space of regular differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g0439903.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g0439904.png" /> (cf. [[Differential form|Differential form]]). The genus of an algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g0439905.png" /> is equal, by definition, to the genus of the complete algebraic curve birationally isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g0439906.png" />. For any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g0439907.png" /> there exists an algebraic curve of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g0439908.png" />. An algebraic curve of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g0439909.png" /> over an algebraically closed field is a [[Rational curve|rational curve]], i.e. it is birationally isomorphic to the projective line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399010.png" />. Curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399011.png" /> (elliptic curves, cf. [[Elliptic curve|Elliptic curve]]) are birationally isomorphic to smooth cubic curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399012.png" />. The algebraic curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399013.png" /> fall into two classes: hyper-elliptic curves and non-hyper-elliptic curves. For non-hyper-elliptic curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399014.png" /> the rational mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399015.png" /> defined by the canonical class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399016.png" /> of the complete smooth curve is an isomorphic imbedding. For a [[Hyper-elliptic curve|hyper-elliptic curve]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399017.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399018.png" /> is a two-sheeted covering of a rational curve, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399019.png" />, ramified at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399020.png" /> points.
+
<!--
 +
g0439901.png
 +
$#A+1 = 38 n = 0
 +
$#C+1 = 38 : ~/encyclopedia/old_files/data/G043/G.0403990 Genus of a curve
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399021.png" /> is a projective plane curve of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399022.png" />, then
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399023.png" /></td> </tr></table>
+
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 $
 +
is equal to the dimension of the space of regular differential  $  1 $-
 +
forms on  $  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 $.
 +
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|rational curve]], i.e. it is birationally isomorphic to the projective line  $  P  ^ {1} $.
 +
Curves of genus  $  g = 1 $(
 +
elliptic curves, cf. [[Elliptic curve|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|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.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399024.png" /> is a non-negative integer measuring the deviation from smoothness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399025.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399026.png" /> has only ordinary double points, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399027.png" /> is equal to the number of singular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399028.png" />. For a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399029.png" /> in space the following estimate is valid:
+
If  $  X $
 +
is a projective plane curve of degree  $  m $,  
 +
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399030.png" /></td> </tr></table>
+
$$
 +
=
 +
\frac{( m - 1 ) ( m - 2 ) }{2}
 +
- d ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399031.png" /> is the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399032.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399033.png" />.
+
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:
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399034.png" /> is the field of complex numbers, then an algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399035.png" /> is the same as a [[Riemann surface|Riemann surface]]. In this case the smooth complex curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399036.png" /> of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399037.png" /> is homeomorphic to the sphere with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043990/g04399038.png" /> handles.
+
$$
 +
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|Riemann surface]]. In this case the smooth complex curve $  X $
 +
of genus g $
 +
is homeomorphic to the sphere with g $
 +
handles.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 {{MR|0092855}} {{ZBL|0078.06602}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 {{MR|0092855}} {{ZBL|0078.06602}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR></table>

Revision as of 19:41, 5 June 2020


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=23842
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article