Adjunction formula

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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[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 YX 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.

Examples[edit]

Degree d hypersurfaces[edit]

Given a smooth degree hypersurface we can compute its canonical and anti-canonical bundles using the adjunction formula. This reads as

which is isomorphic to .

Complete intersections[edit]

For a smooth complete intersection of degrees , the conormal bundle is isomorphic to , so the determinant bundle is and its dual is , showing

This generalizes in the same fashion for all complete intersections.

Curves in a quadric surface[edit]

embeds into 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 . 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

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 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 coordinates z1, ..., 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 df/f. The Poincaré residue is the map that takes the wedge product of a section of ωD 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.

The Canonical Divisor of a Plane Curve[edit]

Let be a smooth plane curve cut out by a degree homogeneous polynomial . We claim that the canonical divisor is where is the hyperplane divisor.

First work in the affine chart . The equation becomes where and . We will explicitly compute the divisor of the differential

At any point either so is a local parameter or so is a local parameter. In both cases the order of vanishing of at the point is zero. Thus all contributions to the divisor are at the line at infinity, .

Now look on the line . Assume that so it suffices to look in the chart with coordinates and . The equation of the curve becomes

Hence

so

with order of vanishing . Hence which agrees with the adjunction formula.

Applications to curves[edit]

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)HdH 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

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 by definition of the bidegree and by bilinearity, so applying Riemann–Roch gives or

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)HdHeH, 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 D1, ..., Dn − 1 of degrees d1, ..., dn − 1 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

In low dimensional topology[edit]

Let S be a complex surface (in particular a 4-dimensional manifold) and let be a smooth (non-singular) connected complex curve. Then[4]

where is the genus of C, denotes the self-intersections and denotes the Kronecker pairing .

See also[edit]

References[edit]

  1. ^ Zhang, Ziyu. "10. Algebraic Surfaces" (PDF). Archived from the original (PDF) on 2020-02-11.
  2. ^ Hartshorne, chapter V, example 1.5.1
  3. ^ Hartshorne, chapter V, example 1.5.2
  4. ^ Gompf, Stipsicz, Theorem 1.4.17