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I need to find divisor of function:

$f(w_0,w_1,w_2)=(w_{0}^{2},w_1w_2)$

on $S=\left \{ w_0w_1-w_2w_3=0 \right \}\subseteq P^3$. And the question is do I need to take into account this surface? There is my solution:

Firstly we will find zeros and poles of f:

  1. $f=0: D=\left \{ w_0=0 \right \}.$

  2. $f= \infty :$ $L_1= \left \{w_1=0 \right \} ; L_2=\left \{ w_2=0 \right \} $

$(f)=mD-n_1L_1-n_2L_2$

Let us take Affine set $A=\left \{ w_3\neq 0 \right \}$ where $x=\frac{w_0}{w_3}, y=\frac{w_1}{w_3}, z=\frac{w_2}{w_3}$. After the intersection of A with 3 curves, we get:

$A\cap D=\left \{ x=0 \right \}; A\cap L_1=\left \{ y=0 \right \};A\cap L_2=\left \{ z=0 \right \}$

hence: $f(x,y,z)=\frac{x^2}{yz}$, so $m=2, n_1=n_2=1$, and $(f)=2D-L_1-L_2$

Is it right? Thanks!

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Dear Mushka, your question doesn't make sense as formulated: a point on $S$ has four homogeneous coordinates and your definition of $f$ only mentions three. Also you will probably want $f$ to be a rational map to $\mathbb P^1$ and not a function. And, yes, you will certainly have to take the equation of $S$ into account. –  Georges Elencwajg Aug 14 '12 at 21:03
    
This is the formulation from my lector, can you please to explain how exactly can I take into account the equation of S? Thank you –  Mushka Aug 17 '12 at 13:42
    
@Georges Elencwajg this is the formulation from my lector, can you please to explain how exactly can I take into account the equation of S? Thank you –  Mushka Aug 18 '12 at 12:36

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