Revision as of 08:24, 14 March 2008 by imported>David Lehavi
In algebraic geometry, the adjunction formula states that if
are smooth algebraic varieties, and
is of codimension 1, then there is a natural isomorphism of sheaves:
examples
- The genus degree formula for plane curves: Let
be a smooth plane curve of degree
. Recall that if
is a line, then
and
. Hence
. Since the degree of
is
, we see that:
- The genus of a curve given by the transversal intersection of two smooth surfaces
: let the degrees of the surfaces be
. Recall that if
is a plane, then
and
. Hence
and therefore
.
e.g. if
are a quadric and a cubic then the degree of the canonical sheaf of the intersection is 6, and so the genus of the interssection curve is 4.
generalizations
proofs
History
references