In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of algebraic geometry, the relative canonical model of a

singular variety 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 ...

of a

mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical ...

where

X

is a particular

canonical The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical examp ...

variety that maps to

X

, which simplifies the structure.

Description

The precise definition is: If

f:Y\to X

is a

resolution Resolution(s) may refer to: Common meanings * Resolution (debate), the statement which is debated in policy debate * Resolution (law), a written motion adopted by a deliberative body * New Year's resolution, a commitment that an individual ma ...

define the adjunction sequence to be the sequence of subsheaves

f_*\omega_Y^;

\omega_X

is invertible

f_*\omega_Y^=I_n\omega_X^

where

I_n

is the higher adjunction ideal. Problem. Is

\oplus_n f_*\omega_Y^

finitely generated? If this is true then

Proj \oplus_n f_*\omega_Y^ \to  X

is called the ''relative canonical model'' of

Y

, or the ''canonical blow-up'' of

X

. Some basic properties were as follows: The relative canonical model was independent of the choice of resolution. Some integer multiple

r

of the canonical divisor of the relative canonical model was Cartier and the number of exceptional components where this agrees with the same multiple of the canonical divisor of Y is also independent of the choice of Y. When it equals the number of components of Y it was called '' crepant''. M. Reid
Canonical 3-folds
(courtesy copy), proceedings of the Angiers 'Journees de Geometrie Algebrique' 1979 It was not known whether relative canonical models were Cohen–Macaulay. Because the relative canonical model is independent of

Y

, most authors simplify the terminology, referring to it as the relative canonical model ''of ''

X

rather than either the relative canonical model ''of ''

Y

or the canonical blow-up of

X

. The class of varieties that are relative canonical models have canonical singularities. Since that time in the 1970s other mathematicians solved affirmatively the problem of whether they are Cohen–Macaulay. The

minimal model program In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible. The subject has its or ...

started by

Shigefumi Mori is a Japanese mathematician, known for his work in algebraic geometry, particularly in relation to the classification of three-folds. Career Mori completed his Ph.D. titled "The Endomorphism Rings of Some Abelian Varieties" under Masayoshi Nagat ...

proved that the sheaf in the definition always is finitely generated and therefore that relative canonical models always exist.

References

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