Assume we have a projective variety $X$ over some algebraically closed field $k$. How can we show that $O_{X}=k$? I tried to do it in simple examples but the proof is not clear to me.
What is a good way to show global sections of projective variety in an algebraic closed field is a constant?
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algebraic-geometry
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0It would be helpful to know: a) WHICH proof you mean and b) WHICH steps in this proof are not clear to you. – 2012-11-20
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0Is $X$ an integral variety ? – 2012-11-20
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0I asked a professor, who answered me "go to affine components, pass to the quotient, take the intersection", and I felt at lost. $X$ is a projective variety in $\mathbb{P}^{n}$. – 2012-11-21
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0@user32240: In this case (if I am right about what a projective variety is in your context), a proof of this fact can be found in Hartshorne "Algebraic Geometry", Ch.1, Theorem 3.4. Does this help? – 2012-11-21
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0This does help. The professor said I can follow Hartshorne, but I thought that must be over my head and did not really try that. – 2012-11-21