adjunction formula

Let 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 $\backslash mathcal\{I\}$. The conormal exact sequence for i is

:$0\; \backslash to\; \backslash mathcal\{I\}/\backslash mathcal\{I\}^2\; \backslash to\; i^*\backslash Omega\_X\; \backslash to\; \backslash Omega\_Y\; \backslash to\; 0,$

where Ω denotes a cotangent bundle. The determinant of this exact sequence is a natural isomorphism

:$\backslash omega\_Y\; =\; i^*\backslash omega\_X\; \backslash otimes\; \backslash operatorname\{det\}(\backslash mathcal\{I\}/\backslash mathcal\{I\}^2)^\backslash vee,$

where $\backslash vee$ denotes the dual of a line bundle.

Suppose that D is a smooth divisor on X. Its normal bundle extends to a line bundle $\backslash mathcal\{O\}(D)$ on X, and the ideal sheaf of D corresponds to its dual $\backslash mathcal\{O\}(-D)$. The conormal bundle $\backslash mathcal\{I\}/\backslash mathcal\{I\}^2$ is $i^*\backslash mathcal\{O\}(-D)$, which, combined with the formula above, gives

:$\backslash omega\_D\; =\; i^*(\backslash omega\_X\; \backslash otimes\; \backslash mathcal\{O\}(D)).$

In terms of canonical classes, this says that

:$K\_D\; =\; (K\_X\; +\; D)|\_D.$

Both of these two formulas are called the adjunction formula.

Given a smooth degree $d$ hypersurface $i:\; X\; \backslash hookrightarrow\; \backslash mathbb\{P\}^n\_S$ we can compute its canonical and anti-canonical bundles using the adjunction formula. This reads as

$\backslash omega\_X\; \backslash cong\; i^*\backslash omega\_\{\backslash mathbb\{P\}^n\}\backslash otimes\; \backslash mathcal\{O\}\_X(d)$

For a smooth complete intersection $i:\; X\; \backslash hookrightarrow\; \backslash mathbb\{P\}^n\_S$ of degrees $(d\_1,\; d\_2)$, the conormal bundle $\backslash mathcal\{I\}/\backslash mathcal\{I\}^2$ is isomorphic to $\backslash mathcal\{O\}(-d\_1)\backslash oplus\; \backslash mathcal\{O\}(-d\_2)$, so the determinant bundle is $\backslash mathcal\{O\}(-d\_1\{-\}d\_2)$ and its dual is $\backslash mathcal\{O\}(d\_1\{+\}d\_2)$, showing

$\backslash omega\_X\; \backslash ,\backslash cong\backslash ,\; \backslash mathcal\{O\}\_X(-n\{-\}1)\backslash otimes\; \backslash mathcal\{O\}\_X(d\_1\{+\}d\_2)$

\,\cong\, \mathcal{O}_X(-n{-}1 {+} d_1 {+} d_2).

$\backslash mathbb\{P\}^1\backslash times\backslash mathbb\{P\}^1$ embeds into $\backslash mathbb\{P\}^3$ as a quadric surface given by the vanishing locus of a quadratic polynomial coming from a non-singular symmetric matrix.{{cite web |last1=Zhang |first1=Ziyu |title=10. Algebraic Surfaces |url=https://ziyuzhang.github.io/ma40188/Lecture19.pdf |archiveurl=https://web.archive.org/web/20200211004951/https://ziyuzhang.github.io/ma40188/Lecture19.pdf|archive-date=2020-02-11 }} We can then restrict our attention to curves on $Y=\; \backslash mathbb\{P\}^1\backslash times\backslash mathbb\{P\}^1$. We can compute the cotangent bundle of $Y$ using the direct sum of the cotangent bundles on each $\backslash mathbb\{P\}^1$, so it is $\backslash mathcal\{O\}(-2,0)\backslash oplus\backslash mathcal\{O\}(0,-2)$. Then, the canonical sheaf is given by $\backslash mathcal\{O\}(-2,-2)$, 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 $f\; \backslash in\; \backslash Gamma(\backslash mathcal\{O\}(a,b))$, can be computed as

:$$

\omega_C \,\cong\, \mathcal{O}(-2,-2)\otimes \mathcal{O}_C(a,b) \,\cong\, \mathcal{O}_C(a{-}2, b{-}2).

{{see also|Poincaré residue}}

The restriction map $\backslash omega\_X\; \backslash otimes\; \backslash mathcal\{O\}(D)\; \backslash to\; \backslash omega\_D$ 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 $\backslash mathcal\{O\}(D)$ 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 map

:$\backslash eta\; \backslash otimes\; \backslash frac\{s\}\{f\}\; \backslash mapsto\; s\backslash frac\{\backslash partial\backslash eta\}\{\backslash partial\; f\}\backslash bigg|\_\{f\; =\; 0\},$

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, {{nowrap begin}}∂f/∂z_i ≠ 0{{nowrap end}}, then this can also be expressed as

:$\backslash frac\{g(z)\backslash ,dz\_1\; \backslash wedge\; \backslash dotsb\; \backslash wedge\; dz\_n\}\{f(z)\}\; \backslash mapsto\; (-1)^\{i-1\}\backslash frac\{g(z)\backslash ,dz\_1\; \backslash wedge\; \backslash dotsb\; \backslash wedge\; \backslash widehat\{dz\_i\}\; \backslash wedge\; \backslash dotsb\; \backslash wedge\; dz\_n\}\{\backslash partial\; f/\backslash partial\; z\_i\}\backslash bigg|\_\{f\; =\; 0\}.$

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

:$\backslash omega\_D\; \backslash otimes\; i^*\backslash mathcal\{O\}(-D)\; =\; i^*\backslash omega\_X.$

On an open set U as before, a section of $i^*\backslash mathcal\{O\}(-D)$ 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 ω_D and a section of $i^*\backslash mathcal\{O\}(-D)$.

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.

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

First work in the affine chart $Z\; \backslash neq\; 0$. 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

:$\backslash omega\; :=\; \backslash frac\{dx\}\{\backslash partial\; f\; /\; \backslash partial\; y\}\; =\; \backslash frac\{-dy\}\{\backslash partial\; f\; /\; \backslash partial\; x\}.$

At any point $(x\_0,\; y\_0)$ either $\backslash partial\; f\; /\; \backslash partial\; y\; \backslash neq\; 0$ so $x\; -\; x\_0$ is a local parameter or

$\backslash partial\; f\; /\; \backslash partial\; x\; \backslash neq\; 0$ so $y\; -\; y\_0$ is a local parameter.

In both cases the order of vanishing of $\backslash omega$ at the point is zero. Thus all contributions to the divisor $\backslash text\{div\}(\backslash omega)$ are at the line at infinity, $Z\; =\; 0$.

Now look on the line $\{Z\; =\; 0\}$. Assume that $[1,\; 0,\; 0]\; \backslash not\backslash in\; C$ so it suffices to look in the chart $Y\; \backslash neq\; 0$ 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

:$\backslash partial\; f/\backslash partial\; x\; =\; y^d\; \backslash frac\{\backslash partial\; g\}\{\backslash partial\; v\}\; \backslash frac\{\backslash partial\; v\}\{\backslash partial\; x\}\; =\; y^\{d-1\}\backslash frac\{\backslash partial\; g\}\{\backslash partial\; v\}$

so

:$\backslash omega\; =\; \backslash frac\{-dy\}\{\backslash partial\; f\; /\; \backslash partial\; x\}\; =\; \backslash frac\{1\}\{u^2\}\; \backslash frac\{du\}\{y^\{d-1\}\backslash partial\; g/\; \backslash partial\; v\}\; =\; u^\{d-3\}\; \backslash frac\{du\}\{\backslash partial\; g\; /\; \backslash partial\; v\}$

with order of vanishing $\backslash nu\_p(\backslash omega)\; =\; (d-3)\backslash nu\_p(u)$. Hence $\backslash text\{div\}(\backslash omega)\; =\; (d-3)[C\; \backslash cap\; \backslash \{Z\; =\; 0\backslash \}]$ which agrees with the adjunction formula.

The genus-degree formula for plane curves can be deduced from the adjunction formula.Hartshorne, chapter V, example 1.5.1 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 {{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

:$g\; =\; \backslash tfrac12(d\{-\}\; 1)(d\; \{-\}\; 2).$

Similarly,Hartshorne, chapter V, example 1.5.2 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 $((d\_1,d\_2),(e\_1,e\_2))\backslash mapsto\; d\_1\; e\_2\; +\; d\_2\; e\_1$ by definition of the bidegree and by bilinearity, so applying Riemann–Roch gives $2g-2\; =\; d\_1(d\_2\{-\}2)\; +\; d\_2(d\_1\{-\}2)$ or

:$g\; =\; (d\_1\; \{-\}\; 1)(d\_2\; \{-\}\; 1)\; \backslash ,=\backslash ,\; d\_1\; d\_2\; -\; d\_1\; -\; d\_2\; +\; 1.$

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 {{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

:$g\; =\; de(d\; +\; e\; -\; 4)\; /\; 2\; +\; 1.$

More generally, if C is the complete intersection of {{math|n − 1}} hypersurfaces {{math|D₁, ..., D_{n − 1}}} of degrees {{math|d₁, ..., d_{n − 1}}} in Pⁿ, then an inductive computation shows that the canonical class of C is $(d\_1\; +\; \backslash cdots\; +\; d\_\{n-1\}\; -\; n\; -\; 1)d\_1\; \backslash cdots\; d\_\{n-1\}\; H^\{n-1\}$. The Riemann–Roch theorem implies that the genus of this curve is

:$g\; =\; 1\; +\; \backslash tfrac\{1\}\{2\}(d\_1\; +\; \backslash cdots\; +\; d\_\{n-1\}\; -\; n\; -\; 1)d\_1\; \backslash cdots\; d\_\{n-1\}.$

Let S be a complex surface (in particular a 4-dimensional manifold) and let $C\backslash to\; S$ be a smooth (non-singular) connected complex curve. ThenGompf, Stipsicz, Theorem 1.4.17

$2g(C)-2=[C]^2-c\_1(S)[C]$

where $g(C)$ is the genus of C, $[C]^2$ denotes the self-intersections and $c\_1(S)[C]$ denotes the Kronecker pairing .

• Logarithmic form
• Poincare residue
• Thom conjecture

{{Reflist}}

• 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.

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Category:Algebraic geometry