arithmetically Cohen-Macaulay.

If $S$ has dimension $d=0$, then geometrically it does not make sense to take a hyperplane section. However in this case the ideal $I_S+(H)/(H)\subseteq R/(H)$ is Artinian (i.e. $R/I_S+(H)$ has Krull-dimension 0). In this case, we will abuse terminology and still call $I_S+(H)/(H)$ the ideal of a general hyperplane section of $S$, whenever $H\in R$ is a general linear form. The short exact sequence relating the ideals of $S$ and of a general hyperplane section is $$0{\longrightarrow}I_S(-1)\stackrel{\cdot H}{{\longrightarrow}} I_S{\longrightarrow}I_S+(H)/(H){\longrightarrow}0.$$ We refer the interested reader to Section 1.3 of [@mi98b] for facts about hyperplane and hypersurface sections.

Lifting the determinantal property, and good determinantal schemes in flat families
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In this note, we address the question of whether it is possible to lift the property of being standard or good determinantal from a general hyperplane section of a scheme to the scheme itself. For schemes of codimension $2$, the Hilbert-Burch Theorem states that being standard determinantal is equivalent to being arithmetically Cohen-Macaulay. So this question is a natural generalization of the questions that were investigated by Huneke and Ulrich in [@hu93], by Migliore in [@mi94], and by the author in [@go06].

Before starting our discussion, we would like to 