In mathematics, the canonical bundle of a non-singular algebraic variety
An important tool of modern birational geometry is inversion of adjunction, which allows one to deduce results about the singularities of
[1] One notes that For example, for the minimal genus 1 fibration of a (quasi)-bielliptic surface induced by the Albanese morphism, the canonical bundle formula gives that this fibration has no multiple fibers.
A similar deduction can be made for any minimal genus 1 fibration of a K3 surface.
On the other hand, a minimal genus one fibration of an Enriques surface will always admit multiple fibers and so, such a surface will not admit a section.
If the canonical class is effective, then it determines a rational map from V into projective space.
The n-canonical map sends V into a projective space of dimension one less than the dimension of the global sections of the nth multiple of the canonical class.
n-canonical maps may have base points, meaning that they are not defined everywhere (i.e., they may not be a morphism of varieties).
A global section of the canonical bundle is therefore the same as an everywhere-regular differential form.
The degree of the canonical class is 2g − 2 for a curve of genus g.[2] Suppose that C is a smooth algebraic curve of genus g. If g is zero, then C is P1, and the canonical class is the class of −2P, where P is any point of C. This follows from the calculus formula d(1/t) = −dt/t2, for example, a meromorphic differential with double pole at the origin on the Riemann sphere.
If C has genus two or more, then the canonical class is big, so the image of any n-canonical map is a curve.
A canonical curve of genus g always sits in a projective space of dimension g − 1.
For example if P is a polynomial of degree 6 (without repeated roots) then is an affine curve representation of a genus 2 curve, necessarily hyperelliptic, and a basis of the differentials of the first kind is given in the same notation by This means that the canonical map is given by homogeneous coordinates [1: x] as a morphism to the projective line.
The rational normal curve for higher genus hyperelliptic curves arises in the same way with higher power monomials in x.
Otherwise, for non-hyperelliptic C which means g is at least 3, the morphism is an isomorphism of C with its image, which has degree 2g − 2.
[3] There is a converse, which is a corollary to the Riemann–Roch theorem: a non-singular curve C of genus g embedded in projective space of dimension g − 1 as a linearly normal curve of degree 2g − 2 is a canonical curve, provided its linear span is the whole space.
In fact the relationship between canonical curves C (in the non-hyperelliptic case of g at least 3), Riemann-Roch, and the theory of special divisors is rather close.
Effective divisors D on C consisting of distinct points have a linear span in the canonical embedding with dimension directly related to that of the linear system in which they move; and with some more discussion this applies also to the case of points with multiplicities.
[4][5] More refined information is available, for larger values of g, but in these cases canonical curves are not generally complete intersections, and the description requires more consideration of commutative algebra.
The field started with Max Noether's theorem: the dimension of the space of quadrics passing through C as embedded as canonical curve is (g − 2)(g − 3)/2.
[6] Petri's theorem, often cited under this name and published in 1923 by Karl Petri (1881–1955), states that for g at least 4 the homogeneous ideal defining the canonical curve is generated by its elements of degree 2, except for the cases of (a) trigonal curves and (b) non-singular plane quintics when g = 6.
In the exceptional cases, the ideal is generated by the elements of degrees 2 and 3.
Historically speaking, this result was largely known before Petri, and has been called the theorem of Babbage-Chisini-Enriques (for Dennis Babbage who completed the proof, Oscar Chisini and Federigo Enriques).
Outside the hyperelliptic cases, Noether proved that (in modern language) the canonical bundle is normally generated: the symmetric powers of the space of sections of the canonical bundle map onto the sections of its tensor powers.
[9] Petri's work actually provided explicit quadratic and cubic generators of the ideal, showing that apart from the exceptions the cubics could be expressed in terms of the quadratics.
These classical results were proved over the complex numbers, but modern discussion shows that the techniques work over fields of any characteristic.
In general, if the ring above is finitely generated, then it is elementary to see that it is the homogeneous coordinate ring of the image of a k-canonical map, where k is any sufficiently divisible positive integer.
The minimal model program proposed that the canonical ring of every smooth or mildly singular projective variety was finitely generated.
One can show that if the canonical divisor K of V is a nef divisor and the self intersection of K is greater than zero, then V will admit a canonical model (more generally, this is true for normal complete Gorenstein algebraic spaces[11]).
[12] A fundamental theorem of Birkar–Cascini–Hacon–McKernan from 2006[13] is that the canonical ring of a smooth or mildly singular projective algebraic variety is finitely generated.