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A numerical invariant of a non-singular projective algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052690/i0526901.png" />, equal to the dimension of its [[Picard variety|Picard variety]]. If the ground field has characteristic zero (or, more general, if the Picard scheme of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052690/i0526902.png" /> is reduced), then the irregularity coincides with the dimension of the first cohomology space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052690/i0526903.png" /> with coefficients in the structure sheaf.
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A numerical invariant of a non-singular projective algebraic variety $X$, equal to the dimension of its [[Picard variety|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052690/i0526905.png" />-th irregularity of a complete linear system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052690/i0526906.png" /> on a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052690/i0526907.png" /> is defined as
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052690/i0526908.png" /></td> </tr></table>
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$$\sigma^i(D)=\dim H^i(X,\mathcal O_X(D)),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052690/i0526909.png" />.
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where $1\leq i\leq\dim X$.
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>

Latest revision as of 14:34, 1 August 2014

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://encyclopediaofmath.org/index.php?title=Irregularity&oldid=11495
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article