= Real conic without real rational points
{title2=$X=V(t_0^2+t_1^2+t_2^2)\subset\mathbb P^2_{\mathbb R}$}
This smooth <projective plane curve> has no real <rational points>: a sum of three real squares is zero only when all three coordinates vanish, which is forbidden in <projective space>. It is therefore not isomorphic over $\mathbb R$ to the <projective line>, which has rational points. Over $\mathbb C$ it becomes a nonsingular conic isomorphic to a projective line. Thus the ground field changes whether a geometric genus-zero curve admits a projective-line parametrization.
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