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

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

## Examples

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

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

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

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

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

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

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

and so ${\displaystyle c(T_{X})=c(i^{*}T_{Y})/N_{X/Y}}$, where ${\displaystyle c}$ 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.