Talk:Canonical ring

Generalization to other bundles

I don't like the redirection of Kodaira dimension to Pluricanonical ring. There is a notion of Kodaira dimension of a line bundle (the rate of growth of $H^{0}(L^{N})$, defined in the same way) introduced (at least heavily used) by Kawamata. It would be better to remove the redirect and write a separate page. -- Tiphareth 07:29, 9 February 2007 (UTC)

I've added a mention that one can generalize the construction and Kodaira dimension to other line bundles; afaik the pluricanonical ring is the classic example of Kodaira dimension (it's what you refer to when you say the Kodaira dimension of a variety, w/o reference to a line bundle), so the redirect doesn't seem illegit. Nbarth 19:38, 26 May 2007 (UTC)

math.AG/0610203

I'm not expert enough to include it, but this page should surely include mention of the results from http://front.math.ucdavis.edu/math.AG/0610203 . "Abstract: We prove that the canonical ring of a smooth projective variety is finitely generated." —The preceding unsigned comment was added by 66.27.69.167 (talk) 08:03, 5 March 2007 (UTC).

This is a certainly a major result. However, as far as I know it hasn't yet appeared in a peer-reviewed journal, so doesn't fulfil WP:ATT. (To be honest, I'm not sure what the status of the arXiv is in this respect. But I'd be very surprised if it qualified as a suitable reference.) But I can check out when it's due to appear. Artie P.S. 09:13, 2 April 2007 (UTC)

Fundamental conjecture of birational geometry

This section seems to me not to conform to general Wikipedia style guidelines, particularly in that it has hyperlinks in the main body. (Also, the above point about the arXiv applies again.) Perhpas it could be reformatted a bit? Artie P.S. 13:37, 19 June 2007 (UTC)