Genus of a curve
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.
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Genus of a curve. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Genus_of_a_curve&oldid=23842