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I have learned that a regular function on a irreducible projective veriety is constant (if the field is algebraically closed).

Now consider $$\varphi:\mathbb{P}^1\rightarrow\mathbb{P}^n,\quad (t_0:t_1)\mapsto(t_0^n:t_0^{n-1}t_1:t_0^{n-2}t_1^2:...:t_1^n).$$

This is a regular function on the irreducible variety $\mathbb{P}^1$ which is not constant.

Where is my mistake?

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    The theorem is that regular functions from irreducible projective varieties into $k$ are constant. Your example is indeed a regular function, but the theorem does not apply. Another example would be the identity map on any projective variety. – 2017-02-23
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    That's it. Thank you! An other example of what? – 2017-02-23
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    Counterexample to your statement "regular function on an irreducible projective variety is constant" is wrong (without knowing regular function really means regular function into $k$ in this case). – 2017-02-23

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