Abstract:
Original King's conjecture from 1997 stated that for any smooth projective
toric variety X there exists a full strong exceptional collection of line
bundles in the derived category of $X$. Later, counterexamples were constructed
to King's conjecture. However, last year, a preprint by Ballard, Berkesch,
Brown et al. (arXiv:2501.00130) appeared, proving a new version of King’s
conjecture. The key change in the formulation of the conjecture is the
replacement of the derived category of $X$ with a larger category, — the Cox
category. The Cox category is obtained by gluing the derived categories of
toric stacks corresponding to the chambers of the secondary fan of $X$; in
particular, the Cox category depends only on the set of rays of the fan. In
this talk, I will discuss toric GIT factors, their variations, and the
secondary fan. Then, following the aforementioned work (arXiv:2501.00130), I
will explain how the new version of King's conjecture is proven.
