= Global regular functions on an irreducible projective variety
{title2=$H^0(X,\mathcal O_X)=k$}
Over an <algebraically closed field>, every <regular function> on a nonempty irreducible <projective variety> is constant. Regard the function as a morphism to $\mathbb P^1$ avoiding infinity. Its graph is closed; the <closedness of projection from projective space> makes its image closed in $\mathbb P^1$. A proper closed subset there is finite, and an irreducible finite image is a singleton. Irreducibility is the usual variety convention; on a disconnected projective algebraic set different components may have different constants.
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