In mathematics, the pluricanonical ring of an

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...

''V'' (which is

non-singular In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In ca ...

), or of a complex manifold, is the

graded ring In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...

R(V,K)=R(V,K_V) \,

of sections of powers of the

canonical bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, it ...

''K''. Its ''n''th graded component (for

n\geq 0

) is: :

R_n := H^0(V, K^n),\

that is, the space of sections of the ''n''-th

tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...

''K''^''n'' of the canonical bundle ''K''. The 0th graded component

R_0

is sections of the trivial bundle, and is one-dimensional as ''V'' is projective. The projective variety defined by this graded ring is called the canonical model of ''V'', and the dimension of the canonical model is called the

Kodaira dimension In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical model of a projective variety ''X''. Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the notation ''κ''. ...

of ''V''. One can define an analogous ring for any line bundle ''L'' over ''V''; the analogous dimension is called the

Iitaka dimension In algebraic geometry, the Iitaka dimension of a line bundle ''L'' on an algebraic variety ''X'' is the dimension of the image of the rational map to projective space determined by ''L''. This is 1 less than the dimension of the section ring of ...

. A line bundle is called big if the Iitaka dimension equals the dimension of the variety.

Properties

Birational invariance

The canonical ring and therefore likewise the Kodaira dimension is a

birational invariant In algebraic geometry, a birational invariant is a property that is preserved under birational equivalence. Formal definition A birational invariant is a quantity or object that is well-defined on a birational equivalence class of algebraic variet ...

: Any birational map between smooth compact complex manifolds induces an isomorphism between the respective canonical rings. As a consequence one can define the Kodaira dimension of a singular space as the Kodaira dimension of a

desingularization In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety ''V'' has a resolution, a non-singular variety ''W'' with a proper birational map ''W''→''V''. For varieties over fields of characterist ...

. Due to the birational invariance this is well defined, i.e., independent of the choice of the desingularization.

Fundamental conjecture of birational geometry

A basic conjecture is that the pluricanonical ring is finitely generated. This is considered a major step in the Mori program. proved this conjecture.

The plurigenera

The dimension :

P_n = h^0(V, K^n) = \operatorname\ H^0(V, K^n)

is the classically defined ''n''-th ''plurigenus'' of ''V''. The pluricanonical divisor

K^n

, via the corresponding

linear system of divisors In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family. These arose first in the f ...

, gives a map to projective space

\mathbf(H^0(V, K^n)) = \mathbf^

, called the ''n''-canonical map. The size of ''R'' is a basic invariant of ''V'', and is called the Kodaira dimension.

Notes

References

* * {{Citation , first1=Phillip , last1=Griffiths , authorlink=Phillip Griffiths , first2=Joe , last2=Harris , author-link2=Joe Harris (mathematician) , title=Principles of Algebraic Geometry , series=Wiley Classics Library , publisher=Wiley Interscience , year=1994 , isbn=0-471-05059-8 , page=573 Algebraic geometry Birational geometry Structures on manifolds