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In algebraic geometry, the adjunction formula states that if $X,Y$ are smooth algebraic varieties, and $X\subset Y$ is of codimension 1, then there is a natural isomorphism of sheaves:

$K_{X}\cong (K_{Y}\otimes {\mathcal {O}}_{Y}(X))|_{X}$ .

## Examples

• The genus-degree formula for plane curves: Let $C\subset \mathrm {P} ^{2}$ be a smooth plane curve of degree $d$ . Recall that if $H\subset \mathbb {P} ^{2}$ is a line, then $\mathrm {Pic} (\mathbb {P} ^{2})=\mathbb {Z} H$ and $K_{\mathbb {P} ^{2}}\equiv -3H$ . Hence

$K_{C}\equiv (-3H+dH)(dH)$ . Since the degree of $K_{C}$ is $2genus(C)-2$ , we see that:

$genus(C)=(d^{2}-3d+2)/2=(d-1)(d-2)/2$ .

• The genus of a curve given by the transversal intersection of two smooth surfaces $S,T\subset \mathbb {P} ^{3}$ : let the degrees of the surfaces be $c,d$ . Recall that if $H\subset \mathbb {P} ^{3}$ is a plane, then $\mathrm {Pic} (\mathbb {P} ^{3})=\mathbb {Z} H$ and $K_{\mathbb {P} ^{3}}\equiv -4H$ . Hence

$K_{S}\equiv (-4H+cH)|_{S}$ and therefore $K_{S\cap T}\equiv (-4H+cH+dH)cHdH=cd(c+d-4)$ .

e.g. if $S,T$ 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 $i:X\to Y$ be a close embedding of smooth varieties, then we have a short exact sequence:

$0\to T_{X}\to i^{*}T_{Y}\to N_{X/Y}\to 0$ ,

and so $c(T_{X})=c(i^{*}T_{Y})/N_{X/Y}$ , where $c$ is the total chern class.