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.
Examples
Degree d hypersurfaces
Given a smooth degree hypersurface we can compute its canonical and anti-canonical bundles using the adjunction formula. This reads aswhich is isomorphic to.
Complete intersections
For a smooth complete intersection of degrees, the conormal bundle is isomorphic to, so the determinant bundle is and its dual is, showingThis generalizes in the same fashion for all complete intersections.
Curves in a quadric surface
embeds into 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. We can compute the cotangent bundle of using the direct sum of the cotangent bundles on each, so it is. Then, the canonical sheaf is given by, 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, can be computed as
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 ωX. 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 coordinatesz1,..., zn such that for some i, ∂f/∂zi ≠ 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. The Poincaré residue is the map that takes the wedge product of a section of ωD and a section of.
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. 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 to C equals the canonical class of C. This restriction is the same as the intersection product restricted to C, and so the degree of the canonical class of C is. By the Riemann–Roch theorem, g − 1 = d − g + 1, which implies the formula Similarly, if C is a smooth curve on the quadric surface P1×P1 with bidegree , since the canonical class of P1×P1 has bidegree, the adjunction formula shows that the canonical class of C is the intersection product of divisors of bidegrees and. The intersection form on P1×P1 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 curveC 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, which is the intersection product of 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, that is, it has degree. By the Riemann–Roch theorem, this implies that the genus of C is More generally, if C is the complete intersection of hypersurfaces of degrees in Pn, then an inductive computation shows that the canonical class of C is. The Riemann–Roch theorem implies that the genus of this curve is
