Pairs of meromorphic vector fields on projective spaces and stability (Q2720354)

Let \(v:{\mathcal O}_{{\mathbf P}^n}(-r)\to T{\mathbf P}^n\), \(r>0\), \(n\geq 3\), be a meromorphic vector field on \({\mathbf P}^n\), let \(Z\) be its zero locus, and let \(M\) denote its cokernel. Assume that \(\dim Z=0\). Then it is proved in the paper that \(M\) is a reflexive simple rank \(n-1\) sheaf, and moreover \(M\) is stable when \(n=3\). Let \(v_1:{\mathcal O}_{{\mathbf P}^3}(a_1)\to T{\mathbf P}^3\) and \(v_2:{\mathcal O}_{{\mathbf P}^3}(a_2)\to T{\mathbf P}^3\), \(a_2<a_1<0\), be meromorphic vector fields such that \(v_1\) vanishes on a scheme \(Z\) with \(\dim(Z)=0\) and \((v_1,v_2)\) are linearly independent on a scheme \(C\) with \(\dim C=1\). It is also shown in the paper that the pair \((C,Z)\) determines the pair \((v_1,v_2)\) up to an automorphism of \({\mathcal O}_{{\mathbf P}^3}(a_1)\oplus{\mathcal O}_{{\mathbf P}^3}(a_2)\).