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