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Difference between revisions of "Genus of a curve"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich,   "Basic algebraic geometry" , Springer (1977) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977) pp. 91</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
  
  
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G. Springer,   "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.A. Griffiths,   J.E. Harris,   "Principles of algebraic geometry" , Wiley (Interscience) (1978)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 {{MR|0092855}} {{ZBL|0078.06602}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR></table>

Revision as of 21:52, 30 March 2012

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.

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
How to Cite This Entry:
Genus of a curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Genus_of_a_curve&oldid=23842
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article