Adjunction formula

Examples

Degree d hypersurfaces

Given a smooth degree ${\displaystyle d}$ hypersurface ${\displaystyle i:X\hookrightarrow {\mathbb{P}}_S^n}$ we can compute its canonical and anti-canonical bundles using the adjunction formula. This reads as

${\displaystyle {\omega }_X\cong i^{\ast }{\omega }_{{\mathbb{P}}^n}\otimes {\mathcal{O}}_X(d)}$

which is isomorphic to ${\displaystyle {\mathcal{O}}_X(-n-1+d)}$.

Complete intersections

For a smooth complete intersection ${\displaystyle i:X\hookrightarrow {\mathbb{P}}_S^n}$ of degrees ${\displaystyle (d_1,d_2)}$, the conormal bundle ${\displaystyle \mathcal{I}/{\mathcal{I}}^2}$ is isomorphic to ${\displaystyle \mathcal{O}(-d_1)\oplus \mathcal{O}(-d_2)}$, so the determinant bundle is ${\displaystyle \mathcal{O}(-d_1-d_2)}$ and its dual is ${\displaystyle \mathcal{O}(d_1+d_2)}$, showing

${\displaystyle {\omega }_X\cong {\mathcal{O}}_X(-n-1)\otimes {\mathcal{O}}_X(d_1+d_2)\cong {\mathcal{O}}_X(-n-1+d_1+d_2).}$

This generalizes in the same fashion for all complete intersections.

Curves in a quadric surface

${\displaystyle {\mathbb{P}}^1\times {\mathbb{P}}^1}$ embeds into ${\displaystyle {\mathbb{P}}^3}$ as a quadric surface given by the vanishing locus of a quadratic polynomial coming from a non-singular symmetric matrix. We can then restrict our attention to curves on ${\displaystyle Y={\mathbb{P}}^1\times {\mathbb{P}}^1}$. We can compute the cotangent bundle of ${\displaystyle Y}$ using the direct sum of the cotangent bundles on each ${\displaystyle {\mathbb{P}}^1}$, so it is ${\displaystyle \mathcal{O}(-2,0)\oplus \mathcal{O}(0,-2)}$. Then, the canonical sheaf is given by ${\displaystyle \mathcal{O}(-2,-2)}$, which can be found using the decomposition of wedges of direct sums of vector bundles. Then, using the adjunction formula, a curve defined by the vanishing locus of a section ${\displaystyle f\in \mathrm{\Gamma }(\mathcal{O}(a,b))}$, can be computed as

${\displaystyle {\omega }_C\cong \mathcal{O}(-2,-2)\otimes {\mathcal{O}}_C(a,b)\cong {\mathcal{O}}_C(a-2,b-2).}$

Poincaré residue

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

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

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

${\displaystyle \frac{g(z)d{z}_1\wedge \cdots \wedge d{z}_n}{f(z)}\mapsto (-1{)}^{i-1}\frac{g(z)d{z}_1\wedge \cdots \wedge \widehat{d{z}_i}\wedge \cdots \wedge d{z}_n}{\mathrm{\partial }f/\mathrm{\partial }{z}_i}{|}_{f=0}.}$

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

${\displaystyle {\omega }_D\otimes i^{\ast }\mathcal{O}(-D)=i^{\ast }{\omega }_X.}$

On an open set U as before, a section of ${\displaystyle i^{\ast }\mathcal{O}(-D)}$ 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 ${\displaystyle i^{\ast }\mathcal{O}(-D)}$.

Inversion of adjunction

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.

The canonical divisor of a plane curve

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

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

At any point ${\displaystyle (x_0,y_0)}$ either ${\displaystyle \mathrm{\partial }f/\mathrm{\partial }y\ne 0}$ so ${\displaystyle x-x_0}$ is a local parameter or ${\displaystyle \mathrm{\partial }f/\mathrm{\partial }x\ne 0}$ 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 ${\displaystyle [1,0,0]\notin C}$ so it suffices to look in the chart ${\displaystyle Y\ne 0}$ 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

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

so

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

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.

Applications to curves

The genus-degree formula for plane curves can be deduced from the adjunction formula. 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

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

Similarly, 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

${\displaystyle g=1+\frac{1}{2}(d_1+\cdots +d_{n-1}-n-1)d_1\cdots d_{n-1}.}$

In low dimensional topology

Let S be a complex surface (in particular a 4-dimensional manifold) and let ${\displaystyle C\to S}$ be a smooth (non-singular) connected complex curve. Then

${\displaystyle 2g(C)-2=[C{]}^2-c_1(S)[C]}$

where ${\displaystyle g(C)}$ is the genus of C, ${\displaystyle [C{]}^2}$ denotes the self-intersections and ${\displaystyle c_1(S)[C]}$ denotes the Kronecker pairing ${\displaystyle ⟨c_1(S),[C]⟩}$.