A generalized Euler sequence over a complete normal variety X is the unique extension of the trivial bundle V otimes O_X by the sheaf of differentials Omega_X, given by the inclusion of a linear space V in Ext^1(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.
We construct a family of McKay quiver representations on the Danilov resolution of the 1/r(1,a,r - a) singularity. It follows that the resolution is the normalization of the coherent component of the moduli space of stable McKay quiver representation
s for a suitable stability condition. We describe explicitly the corresponding chamber of stability conditions for any coprime numbers r, a.
Following Craw, Maclagan, Thomas and Nakamuras work on Hilbert schemes for abelian groups, we give an explicit description of the G-Hilbert scheme for G equal to a cyclic group of order r, acting on C^3 with weights 1,a,r-a. We describe how the combi
natorial properties of the fan of G-Hilbert scheme relates to the Euclidean algorithm.
