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I'm struggling to calculate a divisor of a function $f$. I have tried to follow the examples but I feel like I'm not understanding the steps. For example, when do we divide by a variable or substitute, which affine chart do we work in? Should we work in multiple and sum together? Which points should we use and when? For example how would I go about finding $div(x/z)$ for $C=V(y^2z − x(x − z)(x − 2z))\subset \mathbb{P}^2$.

I have done a few examples but I do not have a solid set of steps to follow when doing it, it felt like lucky guesswork. If anyone could give a step by step process (if there is one) that would be most appreciated. Thanks in advance.

L3582
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  • First determine zeros and poles of $f$ on $C$ and their multiplicities $m_i$ for the zero $z_i$ and $n_i$ for the pole $p_i$($1$ in the example) then compute $\sum m_iz_i-n_ip_i$ – marwalix Apr 14 '21 at 15:18
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    Here's a related question that may be helpful. On the affine open where $z \neq 0$, we can let $u = x/z$ and $v = y/z$, and then your curve is given by $v^2 = u(u-1)(u-2)$. The linked question should help you compute the divisor restricted to this affine open. $C$ has only one point not contained in this affine open, namely $[0:1:0]$. For this point, you can work in the open where $y \neq 0$ and proceed similarly. – Viktor Vaughn Apr 14 '21 at 17:33
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    Thank you so much for the help! I calculated using both the method linked in the related question and the multiplicities method and found that $div(x/z) = 2[0:0:1]-2[0:1:0].$ If it is not too much trouble would you know if this is correct? Thanks once again. – L3582 Apr 15 '21 at 10:56
  • @L3582 That looks right to me. If you think you have it figured out, consider self-answering your question by posting your solution as an answer below. (By the way, I only get a notification if you reply with @.) – Viktor Vaughn Apr 16 '21 at 05:51

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