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
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 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
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
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
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.^{[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 −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
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
References
- Phillip Griffiths; Joe Harris (1994). Principles of Algebraic Geometry. Wiley-Interscience. ISBN 0-471-05059-8.
- 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.