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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.