Suppose C is an algebraic curve (which has singular points) over an algebraically closed field

Jeffery Clements

Jeffery Clements

Answered question

2022-06-21

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.
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?

Answer & Explanation

Angelo Murray

Angelo Murray

Beginner2022-06-22Added 23 answers

You can define the "order" of a regular function at a point x to be the length of the quotient: O X , x / ( f ) (which will be Artinian, so has finite length). The specifics are probably covered in Fulton's "Intersection theory" or Hartshorne's book.
Addition of details: Here is a sketch that clearly defines this: Let's say that your rational function may be described by f / g and f  / g  at the stalk, which I will call RR 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 / f R )  l ( R / g R ) = l ( R / f  R )  l ( R / g  R ) or l ( R / f R ) + l ( R / g  R ) = l ( R / f  R ) + l ( R / g R ).
Since f g  = f  g, we are done by the following general fact:
l ( R / a b R ) = l ( R / a R ) + l ( R / b R )
Hint: look at the sequence
0  a R / a b R  R / a b R  R / a R  0
PS: One can also give a definition by using the normalization of X, Although less attractive, I believe the above statement to be more practical and comprehensible.

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