A free resolution of a symplectic rank variety (Q1372636)

Let \(F\) be a symplectic vector space over an algebraically closed field \(K\) with characteristic different from 2. Write \(Sp(F)\) and \({\mathfrak {sp}}(F)\) for the symplectic group and its Lie algebra, respectively. Let \(Z\) be the subvariety of \({\mathfrak {sp}}(F)\) of all symplectic endomorphisms of \(F\) whose kernel contains an isotropic subspace of dimension 2. The main result of this paper uses the technique of collapsing to describe a free resolution of length 4 of the ideal of regular functions vanishing on \(Z\). In addition, the paper shows that \(Z\) is normal and has rational singularities as well as providing specific generators for the defining ideal of \(Z\).