Resolutions of orthogonal and symplectic analogues of determinantal ideals

We consider orthogonal and symplectic analogues of determinantal varieties (O) over bar (r1,r2). Such varieties simultaneously generalize usual determinantal varieties and rank varieties of symmetric or anti-symmetric matrices. We find (non-minimal) resolutions of the coordinate rings of the varieties (O) over bar (r1,r2). We determine that "nearly all" such varieties are Cohen-Macaulay and for those that are Cohen-Macaulay we calculate the type. Furthermore, we provide a simple characterization for which varieties (O) over bar (r1,r2) are Gorenstein. As an application, we present a class of ideals in k[Hom(E, F)] that are Gorenstein of codimension 4. (c) 2006 Elsevier Inc. All rights reserved.