Genus of a surface
From Encyclopedia of Mathematics
A numerical birational invariant of a two-dimensional algebraic variety defined over an algebraically closed field . There are two different genera — the arithmetic genus and the geometric genus. The geometric genus
of a complete smooth algebraic surface
is equal to
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i.e. to the dimension of the space of regular differential -forms (cf. Differential form) on
. The arithmetic genus
of a complete smooth algebraic surface
is equal to
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The geometric and arithmetic genera of a complete smooth algebraic surface are related by the formula
, where
is the irregularity of
, which is equal to the dimension of the space of regular differential
-forms on
.
References
[1] | I.R., et al. Shafarevich, "Algebraic surfaces" Proc. Steklov Inst. Math. , 75 (1967) Trudy Mat. Inst. Steklov. , 75 (1965) MR1392959 MR1060325 Zbl 0830.00008 Zbl 0733.14015 Zbl 0832.14026 Zbl 0509.14036 Zbl 0492.14024 Zbl 0379.14006 Zbl 0253.14006 Zbl 0154.21001 |
Comments
References
[a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001 |
[a2] | A. van de Ven, "Compact complex surfaces" , Springer (1984) Zbl 0718.14023 |
[a3] | 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 surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Genus_of_a_surface&oldid=47081
Genus of a surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Genus_of_a_surface&oldid=47081
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article