Алгебралық геометриядағы канондық түйін және оның қолданыстары

In mathematics, the canonical bundle of a non singular algebraic variety of dimension over a field is the line bundle , which is the nth exterior power of the cotangent bundle on
Over the complex numbers, it is the determinant bundle of the holomorphic cotangent bundle Equivalently, it is the line bundle of holomorphic n forms on This is the dualising object for Serre duality on It may equally well be considered as an invertible sheaf. The canonical class is the divisor class of a Cartier divisor on giving rise to the canonical bundle — it is an equivalence class for linear equivalence on , and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor − with canonical. The anticanonical bundle is the corresponding inverse bundle When the anticanonical bundle of is ample, is called a Fano variety.

On a singular variety , there are several ways to define the canonical divisor. If the variety is normal, it is smooth in codimension one. In particular, we can define canonical divisor on the smooth locus. This gives us a unique Weil divisor class on It is this class, denoted by that is referred to as the canonical divisor on

Alternately, again on a normal variety , one can consider , the 'th cohomology of the normalized dualizing complex of This sheaf corresponds to a Weil divisor class, which is equal to the divisor class defined above. In the absence of the normality hypothesis, the same result holds if is S2 and Gorenstein in dimension one.

If the canonical class is effective, then it determines a rational map from V into projective space. This map is called the canonical map. The rational map determined by the nth multiple of the canonical class is the n canonical map. 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). They may have positive dimensional fibers, and even if they have zero dimensional fibers, they need not be local analytic isomorphisms.

The best studied case is that of curves. Here, the canonical bundle is the same as the (holomorphic) cotangent bundle. A global section of the canonical bundle is therefore the same as an everywhere regular differential form. Classically, these were called differentials of the first kind. The degree of the canonical class is 2g − 2 for a curve of genus g.

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. In particular, KC and its multiples are not effective. If g is one, then C is an elliptic curve, and KC is the trivial bundle. The global sections of the trivial bundle form a one dimensional vector space, so the n canonical map for any n is the map to a point.

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. Thus for g = 3 the canonical curves (non hyperelliptic case) are quartic plane curves. All non singular plane quartics arise in this way. There is explicit information for the case g = 4, when a canonical curve is an intersection of a quadric and a cubic surface; and for g = 5 when it is an intersection of three quadrics. 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. 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). The terminology is confused, since the result is also called the Noether–Enriques theorem. 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. This implies for instance the generation of the quadratic differentials on such curves by the differentials of the first kind; and this has consequences for the local Torelli theorem. 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. In the exceptional cases the intersection of the quadrics through the canonical curve is respectively a ruled surface and a Veronese surface. These classical results were proved over the complex numbers, but modern discussion shows that the techniques work over fields of any characteristic.