Let X be a normal arithmetically Gorenstein scheme in Pn. 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
Nearby in the stack
. 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
P3
, and the cone over it in
P4
. As an application we obtain a new proof of some theorems of Lesperance about curves in