Genus of a curve

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A numerical invariant of a one-dimensional algebraic variety defined over a field . The genus of a smooth complete algebraic curve is equal to the dimension of the space of regular differential -forms on (cf. Differential form). The genus of an algebraic curve is equal, by definition, to the genus of the complete algebraic curve birationally isomorphic to . For any integer there exists an algebraic curve of genus . An algebraic curve of genus over an algebraically closed field is a rational curve, i.e. it is birationally isomorphic to the projective line . Curves of genus (elliptic curves, cf. Elliptic curve) are birationally isomorphic to smooth cubic curves in . The algebraic curves of genus fall into two classes: hyper-elliptic curves and non-hyper-elliptic curves. For non-hyper-elliptic curves the rational mapping defined by the canonical class of the complete smooth curve is an isomorphic imbedding. For a hyper-elliptic curve the mapping is a two-sheeted covering of a rational curve, , ramified at points.

If is a projective plane curve of degree , then

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

where is the degree of in .

If is the field of complex numbers, then an algebraic curve is the same as a Riemann surface. In this case the smooth complex curve of genus is homeomorphic to the sphere with handles.


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



[a1] G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10
[a2] P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978)
How to Cite This Entry:
Genus of a curve. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article