# Genus of a curve

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