Rank varieties and their resolutions (Q1355544)

Let \(K\) be an algebraically closed field of characteristic zero, \(F\) be a vector space of dimension \(n\) and \(S_\lambda F\) be the Schur module associated to the partition \(\lambda = (\lambda_1 ,\dots, \lambda_m )\), where \(1 \leq m \leq n\). Let \(m \leq r \leq n\). The variety \(X_r\) of tensors of rank \(\leq r\) is defined to be the set of all tensors \(v\) in \(S_\lambda F\) such that there exists a subspace \(F'\) of \(F\) with \(\text{dim } F' = r\) and \(v \in S_\lambda F' \subset S_\lambda F\). When \(\lambda = (2)\) or \(\lambda = (1,1)\) these rank varieties are well-understood. In particular, the minimal free resolution of \({\mathcal O}_{X_r}\) was described by \textit{T. Józefiak}, \textit{P. Pragacz} and \textit{J. Weyman} [Astérisque 87-88, 109-189 (1981; Zbl 0488.14012)]. The current paper uses the approach of this paper to generalize some of their results to rank varieties \(X_r \subset S_\lambda F\) for an arbitrary partition \(\lambda\). The author gives a set of equations defining \(X_r\) set-theoretically and obtains a minimal free resolution of \({\mathcal O}_{X_r}\). This yields the dimension of \(X_r\) and gives that they are normal and Cohen-Macaulay and have rational singularities. The author gives more precise results in the special case of rank varieties in \(S_d F\) for \(d >2\). Finally, the case of rank varieties of skewsymmetric tensors is studied.