Talk:Divisor (algebraic geometry)

Weil divisor

When you say that a Weil divisor is a linear combination, do you mean you take sums with integral or complex coeficients?

16:23, 28 August 2008 (UTC)

Indeed, this should be specified. It's integral coefficients. I have added this. Thanks. RobHar (talk) 17:05, 28 August 2008 (UTC)

I'm no expert on this, but I think the phrase "In general Cartier behave better than Weil divisors when the variety has singular points" should be expanded or clarified. Isn't it true that if all singularities occur in codimension greater than 1, then Weil divisors will "behave" better, since they won't detect these singularities? Of course, this was probably intended to mean that Weil divisors are meaningless and cannot even be defined unless you have regularity in codimension 1. Hilbertthm90 (talk) 19:19, 4 April 2010 (UTC)

Divisors in complex (analytic geometry)

I think it would be really nice if this article presented a broader view of the notion, with maybe a definition in terms less algebraic. I added a definition for the case of Riemann surfaces, but I am no expert in complex manifolds. Examples in this language should be provided, with for instance a definition of the notion of ampleness in this context. --

Marsupilamov — Preceding unsigned comment added by Marsupilamov (talk • contribs) 15:14, 13 March 2011 (UTC)

Cartier divisors and fraction field

This article is not the correct place to discuss in too much detail the subtleties of rational or meromorphic sections. The proper article to edit to that end is Function field of an algebraic variety or even Function field (scheme theory).

Anyway this article should probably focus on divisors in algebraic varieties, and I doubt the issues Kleiman mentions arise in this context (not that I am an expert, but Hartshorne goes rapidly over the definition)

The section about Weil divisors should be made more precise and the injection of CaCl to WeCl described more clearly.

Marsupilamov (talk) 07:27, 17 April 2011 (UTC)

You're right that this is not the right place to discuss the fraction field in too much detail. Consequently I've simplified the article; it discusses the simplest way that is also technically correct.

It is still very important to understand Cartier divisors on singular (even highly singular) varieties. This is necessary for moduli problems. Kleiman's examples are pretty simple, and they show that these apparent pathologies can arise even on nice spaces. I suggest reading his article (it's linked in the present article through its DOI).

I agree that there is a lot of work to do here. Ozob (talk) 13:13, 17 April 2011 (UTC)

I feel that the description of the actual sheaf of rational functions is not immediately relevant to the notion of a Cartier divisor.

Is it not true that we could define directly the sheaf $K* / O*$ without defining the sheaf K ? This would spare one sheafification, right ?

Also, you accidentally (I presume) reverted the section about linear systems, and I took the liberty to unrevert.

Best,

Marsupilamov (talk) 12:37, 18 April 2011 (UTC)

We could say that a Cartier divisor is a section of K_X^*/O^* and nothing more, but I think that does not provide the reader enough context. What we have in the present article duplicates some of the material at the function field article, but I think that the article is much clearer this way. It might be better still, however, not to define K_X but instead to explain why it is related to the more elementary description of Cartier divisors given in the first paragraph of the section. Right now the article just asserts that the two definitions are the same; it does not explain why.

It is true that we could spare one sheafification. Because sheaving is exact, it carries the exact sequence 0 → O^* → K′^* → K′^*/O^* → 0 of presheaves to the exact sequence 0 → O^* → K_X^* → K_X^*/O^* → 0 of sheaves. (In fancy language, it is part of a geometric morphism of topoi.) But I don't see why we should do it that way. The presheaf K′ is of almost no interest, but K_X is very interesting. I would rather we focus our readers' attention on interesting objects.

Also, my removal of content was entirely unintentional. Thank you for reverting. Ozob (talk) 22:02, 18 April 2011 (UTC)