Quadratic generation of ideals for normally generated nonsingular toric varieties

Prove that for every ample line bundle L on a nonsingular toric variety X, normal generation of L implies that the defining ideal I(X,L) is generated by elements of degree two.

Background

Sturmfels formulated this conjecture in 1995 in the setting of projective nonsingular toric varieties. The conjecture asserts that whenever the section ring of an ample line bundle is generated in degree one, all defining relations of the associated toric embedding can be generated by quadratic relations.

The paper establishes the conjecture for a class of nonsingular toric 3-folds, namely certain toric \mathbb{P}^1-bundles over blowups of \mathbb{P}^2 at at most three invariant points or of \mathbb{F}_0 at at most four invariant points. The general statement for arbitrary nonsingular toric varieties is therefore left unresolved by the results presented.