On resolving vector fields (Q1355578)

Let \(V\) be a singular affine irreducible variety over a field \(k\) and \(\mathcal{D}\) an \(\mathcal{O}_{V}\)-module of \(k\)-linear derivations. Suppose that \(I \subset \mathcal{O}_{V}\) is an ideal such that , for the blow-up \(\tilde{V} =BL_{I}(V)\), \(\mathcal{O}_{\tilde{V}}\cdot \mathcal{D}(\mathcal{O}_{\tilde{V}})\) is locally free. Let \(\mathcal{M}_{\mathcal{D}}(f_{1}, \cdots ,f_{m})\) be the ideal generated by \(f_{i}\delta (f_{j}) -f_{j}\delta (f_{i})\), \(f_{i}\in \mathcal{O}_{V}\), \(i,j = 1, \cdots ,m\), for \(\delta\in \mathcal{D}\). We say that the sequence \(f_{1}, \cdots ,f_{m}\) satisfies condition (*) if, for the ideal \(J\) generated by \(f_{1}, \cdots ,f_{m}\), \[ J^{2}\mathcal{M}_{\mathcal{D}}^{2} (f_{1}, \cdots ,f_{m}) +(\mathcal{M}_{\mathcal{D}} (f_{1}, \cdots ,f_{m}))^{2}\mathcal{D}(\mathcal{O}_{V}) \subset(\mathcal{M}_{\mathcal{D}} (f_{1}, \cdots ,f_{m}))^{3}. \] Then, for \(N \gg 0\), \(J = I^{N}\) has a sequence of generators satisfying condition (*). Conversely, if \(f_{1}, \cdots ,f_{m}\) satisfies condition (*), then, for \(J = (f_{1}, \cdots ,f_{m})\), \(I = J \cdot \mathcal{M}_{\mathcal{D}}(f_{1}, \cdots ,f_{m})\) and the blow-up \(\tilde{V} = BL_{I}(V)\), the sheaf \(\mathcal{O}_{\tilde{V}}\cdot \mathcal{D}(\mathcal{O}_{\tilde{V}})\) is locally free.