Derived Categories of Permutahedral and Stellahedral Varieties (2311.04203v1)

Abstract: We utilize the coherent-constructible correspondence to construct full strongly exceptional collections of nef line bundles in the derived category of a toric variety through the combinatorics of constructible sheaves built from polytopes. To show that sequences are full, we build exact complexes of line bundles that categorify the relations in the McMullen polytope algebra. We compute the homomorphisms between certain constructible sheaves on polytopes and use this to reduce the question of exceptionality to showing that certain set differences of polytopes are contractible. As an application of our method, we construct full strongly exceptional collections of nef line bundles for the toric varieties associated to the permutahedron, stellahedron, and the type $B_n$ Coxeter permutahedron. The line bundles in our collections are indexed by base polytopes of loopless Schubert matroids, independence polytopes of all Schubert matroids, and feasible polytopes of loopless Schubert delta matroids, respectively. Our collections satisfy a number of nice properties: First, the quiver with relations that encodes the endomorphism algebra of the tilting sheaf can be described matroid-theoretically as a slight extension of the notion of weak maps and inclusion of matroids; Second, our collections are invariant under the natural symmetries of the corresponding fans; Finally, the induced semi-orthogonal decomposition of the derived categories refines the cuspidal semi-orthogonal decomposition as studied by Castravet and Tevelev. This gives a full strongly exceptional collection of nef line bundles for the cuspidal parts of the derived categories of our varieties indexed by loopless and coloopless Schubert matroids and Schubert delta matroids.