Difference between revisions of "Irregularity"
From Encyclopedia of Mathematics
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| − | A numerical invariant of a non-singular projective algebraic variety | + | {{TEX|done}} |
| + | 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 | + | 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 | + | where $1\leq i\leq\dim X$. |
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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
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