# 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.

## Contents

## Adjunction for smooth varieties[edit]

### Formula for a smooth subvariety[edit]

Let *X* be a smooth algebraic variety or smooth complex manifold and *Y* be a smooth subvariety of *X*. Denote the inclusion map *Y* → *X* by *i* and the ideal sheaf of *Y* in *X* by . The conormal exact sequence for *i* is

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

where denotes the dual of a line bundle.

### The particular case of a smooth divisor[edit]

Suppose that *D* is a smooth divisor on *X*, its normal bundle extends to a line bundle on *X*, and the ideal sheaf of *D* corresponds to its dual . The conormal bundle is , which, combined with the formula above, gives

In terms of canonical classes, this says that

Both of these two formulas are called the **adjunction formula**.

## Poincaré residue[edit]

The restriction map 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 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

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 *z*_{1}, ..., *z*_{n} such that for some *i*, ∂*f*/∂*z*_{i} ≠ 0, then this can also be expressed as

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

On an open set *U* as before, a section of 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 .

## Inversion of adjunction[edit]

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[edit]

The genus-degree formula for plane curves can be deduced from the adjunction formula.^{[1]} Let *C* ⊂ **P**^{2} be a smooth plane curve of degree *d* and genus *g*. Let *H* be the class of a hypersurface in **P**^{2}, that is, the class of a line. The canonical class of **P**^{2} is −3*H*. 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

Similarly,^{[2]} if *C* is a smooth curve on the quadric surface **P**^{1}×**P**^{1} with bidegree (*d*_{1},*d*_{2}) (meaning *d*_{1},*d*_{2} are its intersection degrees with a fiber of each projection to **P**^{1}), since the canonical class of **P**^{1}×**P**^{1} has bidegree (−2,−2), the adjunction formula shows that the canonical class of *C* is the intersection product of divisors of bidegrees (*d*_{1},*d*_{2}) and (*d*_{1}−2,*d*_{2}−2). The intersection form on **P**^{1}×**P**^{1} is by definition of the bidegree and by bilinearity, so applying Riemann–Roch gives or

This gives a simple proof of the existence of curves of any genus as the graph of a function of degree .

The genus of a curve *C* which is the complete intersection of two surfaces *D* and *E* in **P**^{3} 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

More generally, if *C* is the complete intersection of *n* − 1 hypersurfaces *D*_{1}, ..., *D*_{n − 1} of degrees *d*_{1}, ..., *d*_{n − 1} in **P**^{n}, then an inductive computation shows that the canonical class of *C* is . The Riemann–Roch theorem implies that the genus of this curve is

## See also[edit]

## References[edit]

*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.