= 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. Let
$$
I_x=(f_1(x,y),\ldots,f_r(x,y))\subseteq k[y_0,\ldots,y_n].
$$
Then $x\in\pi(Y)$ 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 $N$, so
$$
\boxed{\pi(Y)=\bigcap_{N\ge0}\{x:I_x\not\supseteq\mathfrak m^N\}.}
$$
For 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.
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