= Graded multiplication map for a projective fiber
{title2=$\mu_N:\bigoplus_i S_{N-d_i}\to S_N$}
For homogeneous fiber equations $f_i$ of fixed degrees $d_i$ in $S=k[y_0,\ldots,y_n]$, put $S_j=0$ for $j<0$. The <linear map> $\mu_N((g_i)_i)=\sum_i g_if_i$ has image the degree-$N$ component of their <homogeneous ideal>. Hence it is surjective exactly when that ideal contains $(y_0,\ldots,y_n)^N$: 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>.
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