= Rational-point projection need not be Zariski closed
Over the real numbers, the closed equation $y_0^2=t y_1^2$ in $\mathbb A^1\times\mathbb P^1$ projects on real points to $\{t\ge0\}$, which is not <Zariski closed>. Projective properness concerns scheme or geometric images, not the image of rational points over an arbitrary field. Similarly, $y_0^2+y_1^2$ has no real projective zero, but its <homogeneous ideal> contains no power of the <irrelevant ideal of projective space>, since it has a complex projective zero.
Back to article page
