Local stability chambers for toric tangent bundles

We classify the anticanonical slope stability of the tangent bundles of \(X_{p,q}=\mathbb{P}_{(\mathbb{P}^1)^{p+q}}(\mathcal O\oplus\mathcal O(1^p,-1^q))\), where \(p,q\) are nonnegative integers and \(p+q\ge1\). The tangent bundle is stable when \(p=q\), strictly semistable when \(|p-q|=1\), and unstable otherwise. For \(p=k\) and \(q=k-1\), the germ of the semistable polarization locus at the anticanonical class is real analytically equivalent to \([0,\infty)^k\times\mathbb{R}^k\), with stable locus \((0,\infty)^k\times\mathbb{R}^k\). In particular, the number of independent local chamber boundaries is unbounded. For a smooth projective toric variety whose tangent bundle is semistable at a polarization \(H_0\), we describe the local chamber at \(H_0\) by the order ideals of a finite poset determined by the equal-slope ray subspaces. If each such subspace is spanned by the rays it contains in every maximal cone, nearby stable polarizations exist precisely when the variety is not a nontrivial toric product. Applied to the existing anticanonical census in dimensions three through six, this criterion gives nearby stability in all 31 indecomposable strictly semistable cases, whose local chambers have one, two, or three independent boundaries. We determine locally free, pairwise nonisomorphic Jordan–Hölder factors both for these cases and for the family \(X_{k,k-1}\).