Adjunction formula explained

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

l{I} . The conormal exact sequence for i is0 \to l{I}/ l{I} ^{2} \to i ^{*\Omega} _{X} \to \Omega _{Y} \to 0 , where Ω denotes a cotangent bundle. The determinant of this exact sequence is a natural isomorphism\omega _{Y} = i ^{*\omega} _{X} ⊗ \operatorname{det}( l{I}/ l{I} ^{2)} ^{\vee,} where \vee denotes the dual of a line bundle.

The particular case of a smooth divisor

l{O}(D) on X, and the ideal sheaf of D corresponds to its dual l{O}(-D) . The conormal bundle l{I}/ l{I} ^{2} is i ^{*l{O}(-D)} , which, combined with the formula above, gives\omega _{D} = i ^{*(\omega} _{X} ⊗ l{O}(D)). In terms of canonical classes, this says thatK _{D} = (K _{X} + D)| _{D.} Both of these two formulas are called the adjunction formula.

Examples

Degree d hypersurfaces

Given a smooth degree

d hypersurface i: X \hookrightarrow P ^{n} _{S} we can compute its canonical and anti-canonical bundles using the adjunction formula. This reads as

\omega _{X} \cong i ^{*\omega} _{P ^{n}} ⊗ l{O} _{X(d)}

which is isomorphic to l{O} _{X(-n{-}1{+}d)} .

Complete intersections

For a smooth complete intersection

i: X \hookrightarrow P ^{n} _{S} of degrees (d _{1,} d _{2)} , the conormal bundle l{I}/ l{I} ^{2} is isomorphic to l{O}(-d _{1) ⊕} l{O}(-d _{2)} , so the determinant bundle is l{O}(-d _{1{-}d} _{2)} and its dual is l{O}(d _{1{+}d} _{2)} , showing

\omega _{X} \cong l{O} _{X(-n{-}1) ⊗} l{O} _{X(d} _{1{+}d} _{2)} \cong l{O} _{X(-n{-}1} { + } d _{1} { + } d _{2).}

This generalizes in the same fashion for all complete intersections.

Curves in a quadric surface

P ^{1 x P} ^{1} embeds into P ^{3} 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 Y = P ^{1 x P} ^{1} . We can compute the cotangent bundle of Y using the direct sum of the cotangent bundles on each P ^{1} , so it is l{O}(-2,0) ⊕ l{O}(0,-2) . Then, the canonical sheaf is given by l{O}(-2,-2) , 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 f \in \Gamma( l{O}(a,b)) , can be computed as\omega _{C} \cong l{O}(-2,-2) ⊗ l{O} _{C(a,b)} \cong l{O} _{C(a{-}2,} b{ - } 2 ).

Poincaré residue

See also: Poincaré residue. The restriction map

\omega _{X} ⊗ l{O}(D) \to \omega _{D} 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 l{O}(D) 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 mapη ⊗ (s)/(f) \mapsto s (\partial η)/(\partial f) | _{f} , 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, ..., z_n such that for some i, ∂f/∂z_i ≠ 0, then this can also be expressed as(g(z)dz _{1} \wedge ... b \wedge dz _{n})/(f(z)) \mapsto ( -1 ) ^{i-1} (g(z)dz _{1} \wedge ... b \wedge \widehat{dz _{i})/(\wedge) ... b \wedge dz _{n}{\partial} f/\partial z _{i}|} _{f} .

Another way of viewing Poincaré residue first reinterprets the adjunction formula as an isomorphism

\omega _{D} ⊗ i ^{*l{O}(-D)=} i ^{*\omega} _{X.} On an open set U as before, a section of i ^{*l{O}(-D)} 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 i ^{*l{O}(-D)} .

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

C \subset P ^{2} be a smooth plane curve cut out by a degree d homogeneous polynomial F(X, Y, Z) . We claim that the canonical divisor is K = (d -3 )[C \cap H] where H is the hyperplane divisor.

First work in the affine chart

Z ≠ 0 . The equation becomes f(x, y) = F(x, y, 1 ) = 0 where x = X/Z and y = Y/Z .We will explicitly compute the divisor of the differential\omega : = (dx)/(\partial f / \partial y) = (- dy)/(\partial f / \partial x) .

At any point

(x _{0,} y _{0)} either \partial f / \partial y ≠ 0 so x - x _{0} is a local parameter or\partial f / \partial x ≠ 0 so y - y _{0} is a local parameter.In both cases the order of vanishing of \omega at the point is zero. Thus all contributions to the divisor div (\omega) are at the line at infinity, Z = 0 .

Now look on the line

{Z = 0 } . Assume that [ 1 , 0 , 0 ] \not\in C so it suffices to look in the chart Y ≠ 0 with coordinates u = 1 /y and v = x/y . The equation of the curve becomes g(u, v) = F(v, 1 , u) = F(x/y, 1 , 1 /y) = y ^{-d} F(x, y, 1 ) = y ^{-d} f(x, y).

Hence

\partial f/\partial x = y ^{d} (\partial g)/(\partial v) (\partial v)/(\partial x) = y ^{d-1} (\partial g)/(\partial v)

so

\omega = (- dy)/(\partial f / \partial x) = (1)/(u ^{2}) (du)/(y ^{d-1} \partial g/ \partial v) = u ^{d-3} (dy)/(\partial g / \partial v)

with order of vanishing

\nu _{p(\omega)} = (d -3 )\nu _{p(u)} . Hence div (\omega) = (d -3 )[C \cap \{Z = 0 \}] which agrees with the adjunction formula.

Applications to curves

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 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

g = \tfrac 12 (d{ - } 1 )(d { - } 2 ).

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

((d _{1,d} _{2),(e} _{1,e} _{2))\mapsto} d _{1} e _{2} + d _{2} e _{1} by definition of the bidegree and by bilinearity, so applying Riemann–Roch gives 2 g -2 = d _{1(d} _{2{-}2)} + d _{2(d} _{1{-}2)} org = (d _{1} { - } 1 )(d _{2} { - } 1 ) = d _{1} d _{2} - d _{1} - d _{2} + 1 .

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

Notes and References

1. Web site: Zhang . Ziyu . 10. Algebraic Surfaces . https://web.archive.org/web/20200211004951/https://ziyuzhang.github.io/ma40188/Lecture19.pdf. 2020-02-11 .
2. Hartshorne, chapter V, example 1.5.1
3. Hartshorne, chapter V, example 1.5.2