# Adjunction formula

In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedded in well-behaved spaces such as projective space or to prove theorems by induction.

## Adjunction for smooth varieties

### Formula for a smooth subvariety

Let X be a smooth algebraic variety or smooth complex manifold and Y be a smooth subvariety of X. Denote the inclusion map YX by i and the ideal sheaf of Y in X by ${\displaystyle {\mathcal {I}}}$ . The conormal exact sequence for i is

${\displaystyle 0\to {\mathcal {I}}/{\mathcal {I}}^{2}\to i^{*}\Omega _{X}\to \Omega _{Y}\to 0,}$

where Ω denotes a cotangent bundle. The determinant of this exact sequence is a natural isomorphism

${\displaystyle \omega _{Y}=i^{*}\omega _{X}\otimes \operatorname {det} ({\mathcal {I}}/{\mathcal {I}}^{2})^{\vee },}$

where ${\displaystyle \vee }$  denotes the dual of a line bundle.

### The particular case of a smooth divisor

Suppose that D is a smooth divisor on X. Its normal bundle extends to a line bundle ${\displaystyle {\mathcal {O}}(D)}$  on X, and the ideal sheaf of D corresponds to its dual ${\displaystyle {\mathcal {O}}(-D)}$ . The conormal bundle ${\displaystyle {\mathcal {I}}/{\mathcal {I}}^{2}}$  is ${\displaystyle i^{*}{\mathcal {O}}(-D)}$ , which, combined with the formula above, gives

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

In terms of canonical classes, this says that

${\displaystyle K_{D}=(K_{X}+D)|_{D}.}$

Both of these two formulas are called the 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 conical and anti-canonical bundles using the adjunction formula. This reads as

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

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.[1]. 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 ${\displaystyle f\in \Gamma ({\mathcal {O}}(a,b))}$ , 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 ωX. 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, ..., zn such that for some i, f/∂zi ≠ 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 ωD and a section of ${\displaystyle i^{*}{\mathcal {O}}(-D)}$ .

## 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.

## Applications to curves

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 hypersurface 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)dg + 1, which implies the formula

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

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

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

This gives a simple proof of the existence of curves of any genus as the graph of a function of degree ${\displaystyle g+1}$ .

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, ..., Dn − 1 of degrees d1, ..., dn − 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

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

## References

1. ^ Zhang, Ziyu. "10. Algebraic Surfaces" (PDF). Archived from the original (PDF) on |archive-url= requires |archive-date= (help).
2. ^ Hartshorne, chapter V, example 1.5.1
3. ^ Hartshorne, chapter V, example 1.5.2