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I'd like to write about the fundamenal theorems about the classification of algebraic varieties.--Enyokoyama (talk) 15:11, 16 June 2013 (UTC)
- Please do. Wikipedia's coverage of birational geometry could be much better. Ozob (talk) 16:09, 16 June 2013 (UTC)
add the additional formula called Iitaka conjecuture
editI add a chapter titled "Iitaka conjecture," which had been a big motivation for the classification theory of algebraic varieties. I'd like to improve this chapter for someone professional.--Enyokoyama (talk) 15:03, 4 November 2013 (UTC)
On κ=-1
editDear Ozob. As mentioned in the article "Kodaira dimension,"
- ... we have to define the Kodaira dimension to be −∞ rather than −1, in order to make the formula κ(X × Y) = κ(X) + κ(Y) true in all cases. For example, the Kodaira dimension of P1 × X is −∞ for all varieties X. That convention is also essential in the statement of the Iitaka conjecture.
Particularlly, in the case of higher dimensions, we have to define κ=−∞. In the literature it is sure that κ=-1 in the past, for example, in
- I.Shafarevich, Algebraic surfaces (original in Russian but a translation into English was published by AMS.)
and
- Robin Hartshorne Algebraic Geometry In Ch V.6 Classification of Surfaces.
Though both of them are admirable textbooks they deal only with algebraic surfaces. And first of all, who named κ "Kodaira dimension" is nobody but Shigeru Iitaka as in the title of THIS article.
The spirit of Kodaira dimension, that is enlarged from the Itallian school of Algebraic Geometry and the idea of genera in topology, gives nice spectacles to the classification of algebraic varieties using fiber structures and additive formula.
In 1995 Shafarevich introduced Kodaira dimension κ=−∞ in
- Igor Shafarevich "On some arithmetic properties of algebraic varieties" at Proceedings of the second Asian mathematical Conference.
Therefore I will rewrite the corresponding part in the article of "Kodaira dimension" without deleting two most textbooks as references. I think that κ=-1 WAS in the past.--Enyokoyama (talk) 02:26, 20 April 2014 (UTC)
- I agree that κ = −∞ is a better definition than κ = −1, but I think a better definition is to say that it is negative. It's not possible to give the Iitaka dimension all the right properties in this case. For example, the dimension of Proj R(X, L) should be dim Spec R(X, L) − 1, which is −1. Because of NPOV, Wikipedia should mention all the possible definitions. Ozob (talk) 03:57, 20 April 2014 (UTC)