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Help me please with this question:

Find divisor of rational function $f=(w_0,w_1)$ on a surface $X=\left \{ w_0w_1-w_2w_3 \right \}\subset \mathbb{P}^{3}$

How should I to take into account this surface?

Thanks a lot!

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1 Answer 1

up vote 6 down vote accepted

I interpret your question as asking about the rational function $f=w_1 /w_0$ on the surface $X$.

The divisor of zeros is $div(f)_0=D_2+D_3$ where $D_2$ is the line $w_1=w_2=0$ and $D_3$ is the line $w_1=w_3=0$.
This can be checked on the affine chart $w_0=1$.

The divisor of poles is $div(f)_\infty=E_2+E_3$ where $E_2$ is the line $w_0=w_2=0$ and $E_3$ is the line $w_0=w_3=0$.
This can be checked on the affine chart $w_1=1$.

The locus of indeterminacy of $f$ consists of the two points $D_3\cap E_3=[0:0:1:0]$ and $D_2\cap E_2=[0:0:0:1]$

And finally the answer to your question is $$ div(f)=div(f)_0-div(f)_\infty=D_2+D_3-E_2-E_3 $$

A picture
The lines $D_2,E_3$ are in one of the two rulings of the quadric $X$ and the lines $D_3,E_2$ on the other one. You might try to draw these four lines as the sides of a square and see where the indeterminacy points lie.

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