Assume that a general hyperplane section of $V$ is a standard determinantal subscheme of ${\mathbb{P}}^n$ with degree matrix ${\mathcal U}=(u_{ji})_{i=1,\ldots,t;\; j=1,\ldots,t+c-1}.$ If either $dim V\geq 2$ or $$u_{1,t}+\cdots+u_{c-1,t}\geq n+1$$ then $V$ is arithmetically Cohen-Macaulay.

If $dim(V)\geq 2$ and a general hyperplane section of $V$ is arithmetically Cohen-Macaulay, then $V$ is arithmetically Cohen-Macaulay (see Proposition 2.1 in [@hu93]). We can then reduce to the case when $V$ is one-dimensional. Let $H$ be a general hyperplane, and let $Z=V\cap H$. From Theorem 3.16 of [@hu93] it follows that the minimum degree $b$ of a minimal generator of $I_{Z|H}$ that is not the image of a minimal generator of $I_V$ under the standard projection $I_V\stackrel{\pi}{{\longrightarrow}} I_{Z|H}$ is $$b\geq u_{1,1}+\cdots+u_{t,t}+u_{t+1,t}+\cdots+u_{t+c-1,t}-n=$$ $$=u_{1,t}+\cdots+u_{c-1,t}+u_{c,1}+u_{c+1,2}+\cdots+u_{t+c-1,t}-n
\geq u_{c,1}+u_{c+1,2}+\cdots+u_{t+c-1,t}+1.$$ In particular, it is bigger than the maximum $u_{c,1}+u_{c+1,2}+\cdots+u_{t+c-1,t}$ of the degrees of the minimal generators of $I_{Z|H}$. Then all the minimal generators of $I_{Z|H}$ are images of the minimal generators of $I_V$. Hence $H^1_*({\mathcal{I}}_V)=0$, and $V$ is arithmetically Cohen-Macaulay.

As we mentioned, every arithmetically Cohen-Macaulay scheme of codimension $2$ is standard determinantal. So Lemma \[aCM\] gives a sufficient condition to conclude that $V$ is standard determinantal if $codim(V)=2$.

\[dim2\] Let $V$ be a projective scheme. If $dim(V)\geq 2$ and a 