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.
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 by "i" and the ideal sheaf of "Y" in "X" by formula_1. The conormal exact sequence for "i" is
where Ω denotes a cotangent bundle. The determinant of this exact sequence is a natural isomorphism
where formula_4 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 formula_5 on "X", and the ideal sheaf of "D" corresponds to its dual formula_6. The conormal bundle formula_7 is formula_8, 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 formula_11 hypersurface formula_12 we can compute its canonical and anti-canonical bundles using the adjunction formula. This reads asformula_13which is isomorphic to formula_14.
Complete intersections.
For a smooth complete intersection formula_12 of degrees formula_16, the conormal bundle formula_7 is isomorphic to formula_18, so the determinant bundle is formula_19 and its dual is formula_20, showingformula_21This generalizes in the same fashion for all complete intersections.
Curves in a quadric surface.
formula_22 embeds into formula_23 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 formula_24. We can compute the cotangent bundle of formula_25 using the direct sum of the cotangent bundles on each formula_26, so it is formula_27. Then, the canonical sheaf is given by formula_28, 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 formula_29, can be computed as
Poincaré residue.
The restriction map formula_31 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 formula_5 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 formula_8 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 ω"D" and a section of formula_8.
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.
The Canonical Divisor of a Plane Curve.
Let formula_38 be a smooth plane curve cut out by a degree formula_11 homogeneous polynomial formula_40. We claim that the canonical divisor is formula_41 where formula_42 is the hyperplane divisor.
First work in the affine chart formula_43. The equation becomes formula_44 where formula_45 and formula_46.
We will explicitly compute the divisor of the differential
At any point formula_48 either formula_49 so formula_50 is a local parameter or
formula_51 so formula_52 is a local parameter.
In both cases the order of vanishing of formula_53 at the point is zero. Thus all contributions to the divisor formula_54 are at the line at infinity, formula_55.
Now look on the line formula_56. Assume that formula_57 so it suffices to look in the chart formula_58 with coordinates formula_59 and formula_60. The equation of the curve becomes 
Hence 
so 
with order of vanishing formula_64. Hence formula_65 which agrees with the adjunction formula.
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 −3"H". 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"−3)"d" − "g" + 1, which implies the formula
Similarly, if "C" is a smooth curve on the quadric surface P1×P1 with bidegree ("d"1,"d"2) (meaning "d"1,"d"2 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 ("d"1,"d"2) and ("d"1−2,"d"2−2). The intersection form on P1×P1 is formula_67 by definition of the bidegree and by bilinearity, so applying Riemann–Roch gives formula_68 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 , 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 P"n", then an inductive computation shows that the canonical class of "C" is formula_71. The Riemann–Roch theorem implies that the genus of this curve is