# Irregularity

From Encyclopedia of Mathematics

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 is reduced), then the irregularity coincides with the dimension of the first cohomology space with coefficients in the structure sheaf.

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

where .

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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://www.encyclopediaofmath.org/index.php?title=Irregularity&oldid=23871

This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article