Differentials of Cox rings: Jaczewski's theorem revisited

TL;DR

This paper revisits Jaczewski's theorem by analyzing the structure of Cox rings and their associated bundles, establishing conditions under which a variety is toric and relating Cox ring finiteness to bundle properties.

Contribution

It characterizes when a smooth projective variety is toric based on the splitting of a generalized Euler sequence bundle, and relates Cox ring finiteness to the structure of this bundle.

Findings

Variety is toric if the bundle R_Λ splits into line bundles.

The bundle R_Λ is described via the sheaf of differentials on the Cox ring's characteristic space.

Finiteness of sections of R_Λ correlates with the Cox ring being finitely generated.

Abstract

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.