# Adjunction formula

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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.