Adjunction formula

The restriction map ${\displaystyle \omega _{X}\otimes {\mathcal {O}}(D)\to \omega _{D}}$ is called the Poincaré residue. Suppose that X is a complex manifold. Then on sections, the Poincaré residue can be expressed as follows. Fix an open set U on which D is given by the vanishing of a function f. Any section over U of ${\displaystyle {\mathcal {O}}(D)}$ can be written as s/f, where s is a holomorphic function on U. Let η be a section over U of ω_X. The Poincaré residue is the map

${\displaystyle \eta \otimes {\frac {s}{f}}\mapsto s{\frac {\partial \eta }{\partial f}}{\bigg |}_{f=0},}$

that is, it is formed by applying the vector field ∂/∂f to the volume form η, then multiplying by the holomorphic function s. If U admits local coordinates z₁, ..., z_n such that for some i, ∂f/∂z_i ≠ 0, then this can also be expressed as

${\displaystyle {\frac {g(z)\,dz_{1}\wedge \dotsb \wedge dz_{n}}{f(z)}}\mapsto (-1)^{i-1}{\frac {g(z)\,dz_{1}\wedge \dotsb \wedge {\widehat {dz_{i}}}\wedge \dotsb \wedge dz_{n}}{\partial f/\partial z_{i}}}{\bigg |}_{f=0}.}$

Another way of viewing Poincaré residue first reinterprets the adjunction formula as an isomorphism

${\displaystyle \omega _{D}\otimes i^{*}{\mathcal {O}}(-D)=i^{*}\omega _{X}.}$

On an open set U as before, a section of ${\displaystyle i^{*}{\mathcal {O}}(-D)}$ is the product of a holomorphic function s with the form df/f. The Poincaré residue is the map that takes the wedge product of a section of ω_D and a section of ${\displaystyle i^{*}{\mathcal {O}}(-D)}$.

The adjunction formula is false when the conormal exact sequence is not a short exact sequence. However, it is possible to use this failure to relate the singularities of X with the singularities of D. Theorems of this type are called inversion of adjunction. They are an important tool in modern birational geometry.

Let ${\displaystyle C\subset \mathbf {P} ^{2}}$ be a smooth plane curve cut out by a degree ${\displaystyle d}$ homogeneous polynomial ${\displaystyle F(X,Y,Z)}$. We claim that the canonical divisor is ${\displaystyle K=(d-3)[C\cap H]}$ where ${\displaystyle H}$ is the hyperplane divisor.

First work in the affine chart ${\displaystyle Z\neq 0}$. The equation becomes ${\displaystyle f(x,y)=F(x,y,1)=0}$ where ${\displaystyle x=X/Y}$ and ${\displaystyle y=Y/Z}$. We will explicitly compute the divisor of the differential

${\displaystyle \omega$ :={\frac {dx}{\partial f/\partial y}}={\frac {-dy}{\partial f/\partial x}}.}

At any point ${\displaystyle (x_{0},y_{0})}$ either ${\displaystyle \partial f/\partial y\neq 0}$ so ${\displaystyle x-x_{0}}$ is a local parameter or ${\displaystyle \partial f/\partial x\neq 0}$ so ${\displaystyle y-y_{0}}$ is a local parameter. In both cases the order of vanishing of ${\displaystyle \omega }$ at the point is zero. Thus all contributions to the divisor ${\displaystyle {\text{div}}(\omega )}$ are at the line at infinity, ${\displaystyle Z=0}$.

Now look on the line ${\displaystyle {Z=0}}$. Assume that ${\displaystyle [1,0,0]\not \in C}$ so it suffices to look in the chart ${\displaystyle Y\neq 0}$ with coordinates ${\displaystyle u=1/y}$ and ${\displaystyle v=x/y}$. The equation of the curve becomes

${\displaystyle g(u,v)=F(v,1,u)=F(x/y,1,1/y)=y^{-d}F(x,y,1)=y^{-d}f(x,y).}$

Hence

${\displaystyle \partial f/\partial x=y^{d}{\frac {\partial g}{\partial v}}{\frac {\partial v}{\partial x}}=y^{d-1}{\frac {\partial g}{\partial v}}}$

so

${\displaystyle \omega ={\frac {-dy}{\partial f/\partial x}}={\frac {1}{u^{2}}}{\frac {du}{y^{d-1}\partial g/\partial v}}=u^{d-3}{\frac {dy}{\partial g/\partial v}}}$

with order of vanishing ${\displaystyle \nu _{p}(\omega )=(d-3)\nu _{p}(u)}$. Hence ${\displaystyle {\text{div}}(\omega )=(d-3)[C\cap \{Z=0\}]}$ which agrees with the adjunction formula.

The genus-degree formula for plane curves can be deduced from the adjunction formula.[2] Let C ⊂ P² be a smooth plane curve of degree d and genus g. Let H be the class of a hyperplane in P², that is, the class of a line. The canonical class of P² is −3H. Consequently, the adjunction formula says that the restriction of (d − 3)H to C equals the canonical class of C. This restriction is the same as the intersection product (d − 3)H · dH restricted to C, and so the degree of the canonical class of C is d(d−3). By the Riemann–Roch theorem, g − 1 = (d−3)d − g + 1, which implies the formula

${\displaystyle g={\tfrac {1}{2}}(d{-}1)(d{-}2).}$

Similarly,[3] if C is a smooth curve on the quadric surface P¹×P¹ with bidegree (d₁,d₂) (meaning d₁,d₂ are its intersection degrees with a fiber of each projection to P¹), since the canonical class of P¹×P¹ has bidegree (−2,−2), the adjunction formula shows that the canonical class of C is the intersection product of divisors of bidegrees (d₁,d₂) and (d₁−2,d₂−2). The intersection form on P¹×P¹ is ${\displaystyle ((d_{1},d_{2}),(e_{1},e_{2}))\mapsto d_{1}e_{2}+d_{2}e_{1}}$ by definition of the bidegree and by bilinearity, so applying Riemann–Roch gives ${\displaystyle 2g-2=d_{1}(d_{2}{-}2)+d_{2}(d_{1}{-}2)}$ or

${\displaystyle g=(d_{1}{-}1)(d_{2}{-}1)\,=\,d_{1}d_{2}-d_{1}-d_{2}+1.}$

The genus of a curve C which is the complete intersection of two surfaces D and E in P³ can also be computed using the adjunction formula. Suppose that d and e are the degrees of D and E, respectively. Applying the adjunction formula to D shows that its canonical divisor is (d − 4)H|_D, which is the intersection product of (d − 4)H and D. Doing this again with E, which is possible because C is a complete intersection, shows that the canonical divisor C is the product (d + e − 4)H · dH · eH, that is, it has degree de(d + e − 4). By the Riemann–Roch theorem, this implies that the genus of C is

${\displaystyle g=de(d+e-4)/2+1.}$

More generally, if C is the complete intersection of n − 1 hypersurfaces D₁, ..., D_{n − 1} of degrees d₁, ..., d_{n − 1} in Pⁿ, then an inductive computation shows that the canonical class of C is ${\displaystyle (d_{1}+\cdots +d_{n-1}-n-1)d_{1}\cdots d_{n-1}H^{n-1}}$. The Riemann–Roch theorem implies that the genus of this curve is

${\displaystyle g=1+{\tfrac {1}{2}}(d_{1}+\cdots +d_{n-1}-n-1)d_{1}\cdots d_{n-1}.}$

• Logarithmic form
• Poincare residue

• Intersection theory 2nd edition, William Fulton, Springer, ISBN 0-387-98549-2, Example 3.2.12.
• Principles of algebraic geometry, Griffiths and Harris, Wiley classics library, ISBN 0-471-05059-8 pp 146–147.
• Algebraic geometry, Robin Hartshorne, Springer GTM 52, ISBN 0-387-90244-9, Proposition II.8.20.