Suppose $k$ is a algebraically closed field,$\mathbb{P}_k^n$ is the projective space over $k$,given the Zariski topology.A regular function on $\mathbb{P}_k^n$ is a function $f:\mathbb{P}_k^n \longrightarrow k$ which is locally a quotient of two homogeneous polynomials of the same degree with the denominator not vanishing on that open set.$\mathbb{P}_k^n$ replaced with any other variety $X$, we get a regular function on $X$(with little revision to the definition).Then do there exist such a regular function on $\mathbb{P}_k^n$?How to prove or disprove it?Some hints will be much helpful.
Do there exist regular functions on the projective space over a field?
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algebraic-geometry
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3I'm not sure what you're asking. Are you asking whether there are any regular functions on $\mathbf P^n$? It turns out that $\mathscr{O}(Y) \approx k$ (in the obvious way) for any projective (irreducible) variety $Y$. This is Ch. 1, Theorem 3.4(a) in Hartshorne. Or do you mean something else? – 2011-08-08
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5Even without Hartshorne... the constant functions are regular! – 2011-08-08
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0@Dylan Moreland,Yes,that is exactly what I am asking.Thank you for your referrence. – 2011-08-08