Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 21 / 1 / c / Solution

For an affine base, use the homogeneous equations furnished by the question, with fixed degrees $d_i$ in the projective coordinates. At a point $x$, the fiber $Y_x$ is the projective zero set of their specializations; a specialization may be zero, but is still assigned its original degree. LetThen $x\in \pi (Y)$ exactly when $Y_x$ is nonempty. Part (b) says this is equivalent to $I_x$ containing no power of the irrelevant ideal of projective space. In logical form, noncontainment must hold for every $N$, soFor schemes over an arbitrary field, replace $k$ in this fiber calculation by the residue field $\kappa (x)$ and use geometric fiber points. A point belongs to the scheme-theoretic topological image exactly when its fiber is nonempty, which is equivalent to nonemptiness after algebraic closure of that residue field.