# Irregularity

From Encyclopedia of Mathematics

A numerical invariant of a non-singular projective algebraic variety $X$, 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 $1\leq i\leq\dim X$.

#### Comments

#### 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=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