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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.

Outline of proof and generalizations

The outline follows Fulton (see reference below): Let  be a close embedding of smooth varieties, then we have a short exact sequence:

,

and so , where  is the total chern class.

References

• Intersection theory 2nd eddition, William Fulton, Springer, ISBN 0-387-98549-2, Example 3.2.12.
• Prniciples of algebraic geometry, Griffiths and Harris, Wiley classics library, ISBN 0-471-05059-8 pp 146-147.
• Algebraic geomtry, Robin Hartshorn, Springer GTM 52, ISBN 0-387-90244-9, Proposition II.8.20.