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Irregularity

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A numerical invariant of a non-singular projective algebraic variety , equal to the dimension of its Picard variety. If the ground field has characteristic zero (or, more general, if the Picard scheme of X is reduced), then the irregularity coincides with the dimension of the first cohomology space H^1(X,\mathcal O_X) with coefficients in the structure sheaf.

A variety with non-zero irregularity is called irregular, and a variety with zero irregularity — regular. Sometimes the i-th irregularity of a complete linear system |D| on a variety X is defined as

\sigma^i(D)=\dim H^i(X,\mathcal O_X(D)),

where 1\leq i\leq\dim X.


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References

[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
How to Cite This Entry:
Irregularity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irregularity&oldid=32658
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article