= Solution
A <projective variety> is a Zariski-closed <subset> of <projective space> $\mathbb P_k^n$, equivalently the common <zero set> of <homogeneous polynomials>. <Homogeneity> makes their vanishing independent of the nonzero scalar used to represent a projective <point>.
A <morphism of varieties>[morphism] is a map locally given by regular coordinate functions in <affine charts>. Equivalently, near each <point> of a projective source embedded in $\mathbb P^n$, a <morphism of varieties>[morphism] to $\mathbb P^m$ can be written
$$
\boxed{[x]\longmapsto[F_0(x):\cdots:F_m(x)]}
$$
using <homogeneous polynomials> of a common degree that are not all zero on that <neighbourhood>. The expressions must agree on overlaps, and for a specified target <variety> their values must lie in it. A single global tuple is a sufficient description when available, but is not part of the general definition.
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