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

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