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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.

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|Riemann surface]]. In this case the smooth complex curve  $  X $
of genus  $  g $
is homeomorphic to the sphere with  $  g $
handles.

====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>

====Comments====

====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>