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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Pic}(\mathbb{P}^3)=\mathbb{Z}H} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_{\mathbb{P}^3}\equiv -4H} . Hence

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_S\equiv (-4H+cH) |_S} and therefore Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_{S\cap T}\equiv (-4H+cH+dH)cH dH=cd(c+d-4)} .

e.g. if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 generaliztions

The outline follows Fulton (see reference below): Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i:X\to Y} be a close embedding of smooth varieties, then we have a short exact sequence:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\to T_X\to i^* T_Y \to N_{X/Y}\to 0} ,

and so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c(T_X) = c(i^* T_Y)/N_{X/Y}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.