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Do you mean actual divisor corresponding to some rational function? Then look at the zeros and poles (ie. factor the numerator and denominator)denominator, this part might need some commutative algebra in general). Otherwise, as Kevin said divisors are just formal sums of points. They are book-keeping devices that are boring without object that they book-keep for (if this made sense).

I suggest you look at projective curves, where all the nice properties are more obvious (for compactness reasons). For canonical divisors explicitly factor a differential (again the class is canonical, there are many divisors depending on the choice of a differential). The only difference is that now you need the differential part (dx) and it will change your function as you go from chart to chart.

good references for these matters Otherwise computing its divisor is basically the same.

This is the explicit approach taken in Miranda's Algebraic Curves and Riemann Surfaces. If you have good understanding (i don't) of manifolds and bundles you can take the tangent bundle approach.

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Do you mean actual divisor corresponding to some rational function? Then look at the zeros and poles (ie. factor the numerator and denominator). Otherwise, as Kevin said divisors are just formal sums of points. They are book-keeping devices that are boring without object that they book-keep for (if this made sense).

I suggest you look at projective curves, where all the nice properties are more obvious (for compactness reasons). For canonical divisors find divisors of differentials explicitly factor a differential (again the class is canonical, there are many divisors)divisors depending on the choice of a differential). The differential gives only difference is that now you a transformation between need the affine charts differential part (in terms of the original equation of the curve)dx) will change your function as you go from chart to chart.

good references for these matters is Miranda's Algebraic Curves and Riemann Surfaces.

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Do you mean actual divisor corresponding to some rational function? Then look at the zeros and poles (ie. factor the numerator and denominator). Otherwise, as Kevin said divisors are just formal sums of points. They are book-keeping devices that are boring without object that they book-keep for (if this made sense).

I suggest you look at projective curves, where all the nice properties are more obvious. For canonical divisors find divisors of differentials (again the class is canonical, there are many divisors). The differential gives you a transformation between the affine charts (in terms of the original equation of the curve).