Graded multiplication map for a projective fiber (${\mu }_N:{⨁}_i{S}_{N-d_i}\to S_N$)

For homogeneous fiber equations $f_i$ of fixed degrees $d_i$ in $S=k[y_0,\dots ,y_n]$, put $S_j=0$ for $j<0$. The linear map ${\mu }_N((g_i{)}_i)={\sum }_i{g}_i{f}_i$ has image the degree-$N$ component of their homogeneous ideal. Hence it is surjective exactly when that ideal contains $(y_0,\dots ,y_n{)}^N$: the degree-$N$ monomials generate this power. In a family, the matrix entries are regular functions on the base. Failure of surjectivity is therefore a closed determinantal variety, cut out by the maximal-size matrix minors.