词条 | Adjunction formula |
释义 |
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 varietiesFormula for a smooth subvarietyLet X be a smooth algebraic variety or smooth complex manifold and Y be a smooth subvariety of X. Denote the inclusion map {{nowrap|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 divisorSuppose 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{{see also|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 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, {{nowrap begin}}∂f/∂zi ≠ 0{{nowrap end}}, 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 {{nowrap|df/f}}. The Poincaré residue is the map that takes the wedge product of a section of ωD and a section of . Inversion of adjunctionThe 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 curvesThe genus-degree formula for plane curves can be deduced from the adjunction formula.[1] 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 {{nowrap|(d − 3)H}} to C equals the canonical class of C. This restriction is the same as the intersection product {{nowrap|(d − 3)H · dH}} restricted to C, and so the degree of the canonical class of C is {{nowrap|d(d − 3)}}. By the Riemann–Roch theorem, {{nowrap begin}}g − 1 = (d − 3)d − g + 1{{nowrap end}}, which implies the formula Similarly,[2] 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 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 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 {{math|(d − 4)H|D}}, which is the intersection product of {{nowrap|(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 {{math|(d + e − 4)H · dH · eH}}, that is, it has degree {{math|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 {{math|n − 1}} hypersurfaces {{math|D1, ..., Dn − 1}} of degrees {{math|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 See also
References1. ^Hartshorne, chapter V, example 1.5.1 2. ^Hartshorne, chapter V, example 1.5.2
1 : Algebraic geometry |
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