Find divisor of rational function f=(w_0,w_1) on a surface X=\{w_0w_1-w_2w_3\}\subset

beuz89100g6c

beuz89100g6c

Answered question

2022-02-18

Find divisor of rational function f=(w0,w1) on a surface X={w0w1w2w3}PP3
How should I to take into account this surface?

Answer & Explanation

Cypeexorpjng

Cypeexorpjng

Beginner2022-02-19Added 6 answers

I interpret your question as asking about the rational function f=w1w0 on the surface X.
The divisor of zeros is ÷(f)0=D2+D3 where D2 is the line w1=w2=0 and D3 is the line w1=w3=0. This can be checked on the affine chart w0=1.
The divisor of poles is ÷(f)=E2+E3 where E2 is the line w0=w2=0 and E3 is the line w0=w3=0. This can be checked on the affine chart w1=1.
The locus of indeterminacy of f consists of the two points D3E3=[0:0:1:0] and D2E2=[0:0:0:1]
And finally the answer to your question is
÷(f)=÷(f)0÷(f)=D2+D3E2E3
A picture The lines D2,E3 are in one of the two rulings of the quadric X and the lines D3,E2 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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