Adjunction formula

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.

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.