general hyperplane section of $V$ is aCM, then $V$ is aCM. Therefore the graded Betti numbers of $V$ coincide with the graded Betti numbers of a general hyperplane section of $V$ (for more details see [@mi98b], Theorem 1.3.6). Moreover, for a scheme of codimension $2$ the property of being standard determinantal can be decided by checking the graded Betti numbers. In fact, a scheme of codimension $2$ is standard determinantal if and only if it is aCM, if and only if a minimal free resolution of its saturated ideal has length $2$. Hence if $dim(V)\geq 2$ and $codim(V)=2$, we can decide whether $V$ is standard determinantal by looking at the graded Betti numbers of a general hyperplane section. However, if $codim(V)\geq 3$ then the property of being standard determinantal cannot in general be decided by looking at the graded Betti numbers. In other words, there are schemes which are not standard determinantal, but have the same graded Betti numbers as a standard determinantal scheme (see e.g. Example \[vero\]).

In very special cases the graded Betti numbers of a homogeneous ideal $I$ can force the ideal to be standard determinantal, even when the codimension is $3$ or higher. The next is 