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 $Pic(\mathbb {P} ^{2})=\mathbb {Z} H$ 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 $Pic(\mathbb {P} ^{3})=\mathbb {Z} H$ 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.

references

• 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