= Solution
The <Segre embedding> is
$$
\boxed{\sigma_{n,m}([x_0:\cdots:x_n],[y_0:\cdots:y_m])=[x_iy_j]_{0\leq i\leq n,\ 0\leq j\leq m}.}
$$
Rescaling the two representatives multiplies every target coordinate by the same scalar, and at least one <product> is nonzero, so the map is well defined. In affine source charts its coordinate ratios are regular <products>, so it is a <morphism of varieties>[morphism].
Arrange target coordinates into a <matrix> $Z=(z_{ij})$. Every <image> <matrix> has <rank> one and satisfies all homogeneous equations
$$
z_{ij}z_{k\ell}-z_{i\ell}z_{kj}=0.
$$
Conversely, a nonzero <matrix> satisfying these equations has <rank> one. Choose $z_{i_0j_0}\neq0$ and <set> $x_i=z_{ij_0}$, $y_j=z_{i_0j}/z_{i_0j_0}$. The <minors> imply $z_{ij}=x_iy_j$. Therefore
$$
\boxed{\operatorname{im}\sigma_{n,m}=V(\text{all }2\times2\text{ minors}),}
$$
a <projective variety>. The two projective factors are uniquely recoverable from a nonzero column and row. This also shows <injectivity> and supplies the local inverse formulas used in the next part.
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