Gorenstein Biliaison and ACM Sheaves

Let X be a normal arithmetically Gorenstein scheme in P. We give a criterion for all codimension two ACM subschemes of X to be in the same Gorenstein biliaison class on X, in terms of the category of ACM sheaves on X. These are sheaves that correspond to the graded maximal Cohen–Macaulay modules on the homogeneous coordinate ring of X. Using known results on MCM modules, we are able to determine the Gorenstein biliaison classes of codimension two subschemes of certain varieties, including the nonsingular quadric surface in P, and the cone over it in P. As an application we obtain a new proof of some theorems of Lesperance about curves in P, and answer some questions be raised.