Talk:Glossary of classical algebraic geometry

Non-archaic terms
Many terms in this glossary are not archaic at all. A non-exhaustive list is arithmetic genus, geometric genus, generic, linear, quadric, cubic, ..., octic. Should we remove them or specify that they are not archaic? D.Lazard (talk) 07:33, 7 April 2012 (UTC)

Section "Conventions of classical algebraic geometry"
This section has a lot of issues. First it uses "now", "was", ... without defining the corresponding period. Some of the assertions will become correct if adding "until circa 1935" before every occurrence of "was", "were", ... In fact, André Weil's Foundations of algebraic geometry was written at this time for making algebraic geometry rigorous and allowing to consider it over fields of any characteristic. IMO, "algebraic geometry" may be divided in 4 trends. 1/ the old time, before circa 1900 and Hilbert 2/ the beginning of XX-th century, until second world war, where the (sometimes) not rigorous algebraic geometry did coexist with the algebraization of algebraic geometry through commutative algebra 3/ the modern classical algebraic geometry, based on a formalism derived from Weil's foundations and still alive 4/ Scheme theory and its generalization which is being developed in parallel with classical algebraic geometry. Thus every general comment of algebraic geometry is a non sense if the trend to which is applies is not specified.

I have tagged two of the assertions as "dubious". The second item says "[varieties] are now usually considered as abstract varieties" may only be correct if "now" means "in scheme theory", which is wrong, as there are many recent books and papers on algebraic geometry which do not use schemes nor sheaves and dot consider abstract varieties. More, a "Veronese surface" is always considered in the projective 5-space. If it is considered as an abstract variety, it is simply a plane. This is why, when considering abstract varieties one does not talk of "Veronese surface" but of "Veronese embedding" on the plane in the 5-space.

Next item asserts "Varieties were often considered only up to birational isomorphism". This is wp:OR. On the other hand, "singularity theory" is an active (and old) subfield of classical algebraic geometry, which studies properties which are not invariant by birational mappings.

D.Lazard (talk) 17:54, 17 April 2012 (UTC)

"Varieties were often considered only up to birational isomorphism" is not just wp:OR, it's also undefined in this context; there's probably a wikipedia term for the use of words or phrases in an introductory piece that sends the reader off down rabbit-holes trying to define the words used in a definition. whatever, it's unfriendly & unhelpful to newcomers to the subject. this section is supposed to be a glossary. "varieties" of what? where is "isomorphism" defined in the context of algebraic geometry?

duncanrmi (talk) 11:37, 16 August 2016 (UTC)