Relative canonical model
Encyclopedia
In mathematics
, the relative canonical model of a singular variety
is a particular canonical
variety that maps to
, which simplifies the structure. The precise definition is:
If
is a resolution
define the adjunction sequence to be the sequence of subsheaves
if 
is invertible 
where 
is the higher adjunction ideal. Problem. Is 
finitely generated? If this is true then 
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
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
. It was not known whether relative canonical models were Cohen–Macaulay.
Because the relative canonical model is independent of
, most authors simplify the terminology, referring to it as the relative canonical model of 
rather than either the relative canonical model of 
or the canonical blow-up of 
. 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 started by Shigefumi Mori
proved that the sheaf in the definition always is finitely generated and therefore that relative canonical models always exist.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the relative canonical model of a singular variety
is a particular canonical
Canonical
Canonical is an adjective derived from canon. Canon comes from the greek word κανών kanon, "rule" or "measuring stick" , and is used in various meanings....
variety that maps to
, which simplifies the structure. The precise definition is:If
is a resolutionResolution of singularities
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...
define the adjunction sequence to be the sequence of subsheaves
if is invertible where is the higher adjunction ideal. Problem. Is finitely generated? If this is true then 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
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 crepantCrepant resolution
In algebraic geometry, a crepant resolution of a singularity is a resolution that does not affect the canonical class of the manifold. The term "crepant" was coined by by removing the prefix "dis" from the word "discrepant", to indicate that the resolutions have no discrepancy in the canonical...
. It was not known whether relative canonical models were Cohen–Macaulay.
Because the relative canonical model is independent of
, most authors simplify the terminology, referring to it as the relative canonical model of rather than either the relative canonical model of or the canonical blow-up of . 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 started by Shigefumi MoriShigefumi Mori
-References:*Heisuke Hironaka, Fields Medallists Lectures, Michael F. Atiyah , Daniel Iagolnitzer ; World Scientific Publishing, 2007. ISBN 9810231172...
proved that the sheaf in the definition always is finitely generated and therefore that relative canonical models always exist.