Computing in the 🌵 Derived Category 🌵 of a Toric Variety and A King's Conjecture

A major program in algebraic geometry is the study of vector bundles on algebraic varieties, which are analogous to finitely generated projective modules over a ring. In 1956, Grothendieck proved that any vector bundle P^1 splits as a direct sum of line bundles (rank 1 vector bundles). In contrast, for n>3 there are indecomposable vector bundles of rank n-1 on P^n. This is an expository talk about the same questions on a toric variety. I will recall standard facts about representations of quivers and take us on a scenic route for describing the apparatus of exceptional collections for making the bounded derived category of coherent sheaves on P^n explicit. Then, I will explain how this machinery makes it possible to check whether a vector bundle on P^n splits as a direct sum of line bundles. Here are some extra emojis as keywords: 🍩 💘 🌾