King's conjecture and the secondary fan

In 1997 King conjectured that each smooth projective toric variety has a "full strong exceptional collection of line bundles," or a collection which forms a sort of basis for all graded modules over its coordinate ring with respect to its irrelevant ideal.  The conjecture is false, but has motivated recent work in removing the dependence on irrelevant ideal by considering the secondary fan of the toric variety.  I will discuss a version of King's conjecture which my coauthors and I have proven in this case.