$3$-rd hyperplane section by hyperplanes that meet it properly, while its general $3$-rd hyperplane section is not standard determinantal.

The last family of examples that we wish to study consists of non standard determinantal curves, whose general hyperplane section is good determinantal (see Proposition \[n+1curve\]). The result of the next lemma is not new. For completeness we give a simple algebraic proof of it.

\[sqfr\] Let $I\subseteq k[x_0,\ldots,x_n]$ be the ideal generated by all the squarefree monomials of degree $d$. Then $I$ is a good determinantal ideal.

Let $A$ be a matrix of size $d\times (n+1)$ with entries in $k$ such that all the maximal minors of $A$ are nonzero, $A=(a_{i,j})_{1\leq i\leq
  d;\; 0\leq j\leq n}$. Consider the matrix $M$ that we obtain from $A$ by multiplying each entry in the $j$-th column by $x_j$, $M=(a_{i,j}x_j)_{1\leq i\leq
  d;\; 0\leq j\leq n}.$ The minor involving columns $0\leq j_1<\ldots<j_d\leq n$ is $\alpha_{j_1,\ldots,j_d} x_{j_1}\cdot\ldots\cdot x_{j_d}$, where $\alpha_{j_1,\ldots,j_d}$ is the determinant of the submatrix of $A$ consisting of the columns $j_1,\ldots,j_d$. If $d=1$ then $I$ is a complete intersection, hence good determinantal. If $d\geq 2$ the height of $I$ is $n+2-d$, then $I$ is standard determinantal. In particular, $k[x_0,\ldots,x_n]/I$ is Cohen-Macaulay. If we delete 