A generalized Euler sequence over a complete normal variety X is the unique extension of the trivial bundle V ⊗O X by the sheaf of differentials Ω X , given by the inclusion of a linear space V ⊂ Ext 1 X (O X , Ω X ). For Λ, a lattice of Cartier divisors, let R Λ denote the corresponding sheaf associated to V spanned by the first Chern classes of divisors in Λ. We prove that any projective, smooth variety on which the bundle R Λ splits into a direct sum of line bundles is toric. We describe the bundle R Λ 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 Λ and of the Cox ring of Λ. for discussions and remarks. We are also greatly indebt to an anonymous referee who pointed out mistakes in an earlier version of this paper.

Any rational map between affine spaces, projective spaces or toric varieties can be described in terms of their affine, homogeneous, or Cox coordinates. We show an analogous statement in the setting of Mori Dream Spaces. More precisely (in the case of regular maps) we show there exists a finite extension of the Cox ring of the source, such that the regular map lifts to a morphism from the Cox ring of the target to the finite extension. Moreover the extension only involves roots of homogeneous elements. Such a description of the map can be applied in practical computations.addresses: J. Buczyński,

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We construct a family of McKay quiver representations on the Danilov resolution of the 1 r (1, a, r − a) singularity. This allows us to show that the resolution is the normalization of the coherent component of the fine moduli space of θ-stable McKay quiver representations for a suitable stability condition θ. We describe explicitly the corresponding union of chambers of stability conditions for any coprime numbers r, a.