Difference between revisions of "Canonical class"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table> |
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− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Iitaka, "Algebraic geometry, an introduction to birational geometry of algebraic varieties" , Springer (1982) {{MR|}} {{ZBL|0491.14006}} </TD></TR></table> |
Revision as of 21:50, 30 March 2012
The class of divisors, with respect to linear equivalence on an algebraic variety
, which are divisors of differential forms
of maximal degree. If
is a non-singular algebraic variety and
, then in local coordinates
a form
can be written as
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The divisor of
is locally equal to the divisor
of this rational function
. This construction does not depend on the choice of local coordinates and gives the divisor
of
on all of
. Since for any other form
of the same degree as
,
, it follows that
, and corresponding divisors are equivalent. The canonical class
thus constructed is the first Chern class of the sheaf
of regular differential forms of degree
. Its numerical characteristics (degree, index, self-intersections, etc.) are effectively calculable invariants of the algebraic variety.
If is a non-singular projective curve of genus
, then
. For elliptic curves and, more generally, for Abelian varieties,
. If
is a non-singular hypersurface of degree
in projective space
, then
, where
is a hyperplane section of it.
See also Canonical imbedding.
References
[1] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 |
Comments
References
[a1] | S. Iitaka, "Algebraic geometry, an introduction to birational geometry of algebraic varieties" , Springer (1982) Zbl 0491.14006 |
Canonical class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Canonical_class&oldid=23774