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.

Suppose $C$ is an algebraic curve (which has singular points) over an algebraically closed field $k$, and that $f$ is a rational function on $C$. How does one defines the Weil divisor of $f$?

The problem is that the local rings of $C$ at singular points are not DVR's, so I do not have an obvious candidate for an order at a point.

Thanks!

Edit: Let me give an example, inspired by an answer from below. Suppose $C$ is curve $y^2=x^3$. What would be the order of the rational function $x/y$ at the origin?

share|improve this question
add comment

1 Answer

up vote 3 down vote accepted

You can define the "order" of a regular function at a point $x$ to be the length of the quotient: $\mathcal O_{X,x}/(f)$ (which will be Artinian, so has finite length). You can probably find the details in Hartshorne's book or Fulton's "Intersection theory".

Details added: here is a sketch that this is well-defined: Suppose your rational function can be represented by $f/g $ and $f'/g'$ at the stalk, which I will call $R$. $R$ is one-dimensional domain (you can assume less, but let's make it simple). We know $f/g =f'/g'$ and want to prove $l(R/fR) - l(R/gR) = l(R/f'R) - l(R/g'R)$ or $$l(R/fR) + l(R/g'R) = l(R/f'R) + l(R/gR)$$

Since $fg' = f'g$, we are done by the following general fact:

$$l(R/abR) = l(R/aR) + l(R/bR) $$

Hint: look at the sequence $0 \to aR/abR \to R/abR \to R/aR \to 0$

PS: One can also give a definition by using the normalization of $X$, but I think the above is more down-to-earth and computable, if less sexy.

share|improve this answer
    
But by writing this you assume that $f \in \mathcal O_{X,x}$, which need not be the case. Is $\mathcal O_{X,x}$ a valuation ring so we can assume that either $f$ or $1/f$ is in it? –  anonymous Mar 3 '11 at 8:12
    
For example, let $C$ be $Y^2=X^3$. Looking at the function $x/y$ at the origin, what is it order there? I can calculate the order of $x$ and $y$ and substract, but is this well defined and does not depend on the choice of polynomials representing the function near the origin? –  anonymous Mar 3 '11 at 8:46
    
@anonymous: please see the edit. –  curious Mar 3 '11 at 15:17
add comment

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.