Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm trying to understand the relationship between Weil divisors and Cartier divisors, and I would like to see why these are the same in the simple case where $X$ is a nonsingular projective curve over an algebraically closed field. What is the most concrete way of explaining the equivalence between the two sorts of divisors in this situation? In particular, if $P \in X$ is a closed point thought of as a prime Weil divisor, what is the Cartier divisor corresponding to $P$? What about the invertible sheaf corresponding to $P$?

share|improve this question
    
I can only suggest Hartshorne's Algebraic Geometry, Chapter 2 Paragraph 6. But I suppose you read it before! Waiting with you for a different answer... –  Giovanni De Gaetano Dec 20 '11 at 18:38
    
This is explained in Hartshorne: see Chapter II, Proposition 6.11. The idea is that a Cartier divisor is essentially a meromorphic function modulo local regular functions, and so can be thought of as its zero/singular locus with multiplicities attached. –  Zhen Lin Dec 20 '11 at 18:42
    
Yes, this question occurred to me while reading Hartshorne. But his explanation is not so clear, and I got confused when trying to specialize to the case of a smooth curve. –  Justin Campbell Dec 20 '11 at 20:55

1 Answer 1

up vote 8 down vote accepted

1) The Cartier divisor corresponding to the Weil divisor $1.P$ is given by the pair $s=\lbrace (U,z),(V, 1) \rbrace$ described as follows:
$\bullet$ $U$ is an open neigbourhood of $P$ and $z$ is a regular function on $U$ whose sole zero is $P$ with multiplicity one.
$\bullet \bullet$ $V=X\setminus \lbrace P\rbrace $ and of course $1$ is the constant function equal to $1$ on $V$.
(This pair determines a section $s\in \Gamma(X,\mathcal K^* _X/\mathcal O^*_X)$ if you unravel what it means to be a section of a quotient sheaf.)

2) The invertible sheaf corresponding to $P$ is the sheaf denoted by $\mathcal O(P)$.
Its $k$-vector space of sections $\Gamma(W,\mathcal O(P))$ over an open subset $W\subset X$ consists of those rational functions $f\in Rat(W)=Rat(X)$ regular on $W$ except perhaps at $P$, where $f$ is allowed a pole of order at most $1$.

share|improve this answer
    
Thanks, very illuminating! To be clear: such a regular function $z$ in the first bullet point is obtained by lifting a uniformizer in the local ring at $P$ to some open neighborhood, then discarding any other points where it may happen to have a zero? –  Justin Campbell Dec 20 '11 at 20:56
    
Dear Justin: that's it, exactly! –  Georges Elencwajg Dec 20 '11 at 21:32

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.