a general section which is good determinantal. In other words, whether a section of $V$ by $r$ generic hyperplanes is good determinantal for some $r\leq {t+2\choose 2}-3$. We prove that the schemes of the following proposition have a (special) ${ {{t} \choose {2}} }$-th proper hyperplane section which is good determinantal.

\[symm\] Let $X$ be a symmetric matrix of indeterminates of size $(t+1)\times(t+1)$, $t\geq 2$ $$X=\left(\begin{array}{ccccc} x_{0,0} & x_{0,1} & \cdots & \cdots & x_{0,t} \\
x_{0,1} & x_{1,1} & \cdots & \cdots & x_{1,t} \\
\vdots & \vdots & & & \vdots \\
x_{0,t} & x_{1,t} & \cdots & \cdots & x_{t,t} \end{array}\right).$$ Let $V\subseteq{\mathbb{P}}^{{ {{t+2} \choose {2}} }-1}$ be the scheme corresponding to the saturated ideal $I_V=I_t(X)\subseteq R=k[\; x_{i,j}\; |\; 0\leq i\leq j\leq t\;]$, generated by the submaximal minors of $X$.

1.  $V$ is an arithmetically Cohen-Macaulay, integral scheme of codimension $3$ which is not standard determinantal, but every Artinian reduction of its homogeneous coordinate ring is good determinantal.

2.  Let $D$ be a general ${ {{t} \choose {2}} }$-th hyperplane section of $V$. Then $V$ has a proper ${ {{t} \choose {2}} }$-th hyperplane section $C$ that is a good determinantal scheme, and there is a flat 