Geometric genus

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A numerical invariant of non-singular algebraic varieties. In the case of algebraic curves the geometric genus becomes identical with the genus of the curve (cf. Genus of a curve). The geometric genus for algebraic surfaces was first defined from different points of view by A. Clebsch and M. Noether in the second half of the 19th century. Noether also demonstrated the birational invariance of the geometric genus. The geometric genus of a non-singular projective algebraic variety $X$ over an algebraically closed field $k$ is, by definition, the dimension of the space of regular differential forms (cf. Differential form) of degree $n=\dim X$. In such a case the geometric genus of $X$ is denoted by $p_g(X)$. In accordance with Serre's duality theorem

$$p_g(X)=\dim_kH^n(X,\mathcal O_X),$$

where $\mathcal O_X$ is the structure sheaf of $X$. The number $p_g(X)-1$ coincides with the dimension of the canonical system of $X$ (cf. also Divisor). The geometric genus plays an important role in the criterium of rationality of algebraic surfaces (cf. Rational surface) and also in the general classification of algebraic surfaces. The geometric genera of birationally-isomorphic smooth projective varieties coincide.


[1] M. Baldassarri, "Algebraic varieties" , Springer (1956) MR0082172 Zbl 0995.14003 Zbl 0075.15902
[2] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001


See also Arithmetic genus.


[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
How to Cite This Entry:
Geometric genus. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article