Another view on Cox rings: Jaczewski's theorem revisited

I will report on a joint work with Oskar Kedzierski. A generalized Euler sequence over a complete normal variety $X$ is the unique extension of the trivial bundle $V\otimes\mathcal{O}_X$ by the sheaf of differentials $\Omega_X$, given by the inclusion of a linear space $V\subset\mathrm{Ext}^1(\mathcal{O}_X,\Omega_X)$. For $\Lambda$, a lattice of Cartier divisors, let $R_{\Lambda}$ denote the corresponding sheaf associated to $V$ spanned by the first Chern classes of divisors in $\Lambda$. We prove that any projective, smooth variety on which the bundle $R_{\Lambda}$ splits into a direct sum of line bundles is toric. We describe the bundle $R_{\Lambda}$ in terms of the sheaf of differentials on the characteristic space of the Cox ring, provided it is finitely generated. Moreover, we relate the finiteness of the module of sections of $R_{\Lambda}$ and of the Cox ring of $\Lambda$.