Markus Perling

Exceptional sequences of invertible sheaves on rational surfaces

In this article we consider exceptional sequences of invertible sheaves on smooth complete rational surfaces. We show that to every such sequence one can associate a smooth complete toric surface in a canonical way. We use this structural result to prove various theorems on exceptional and strongly exceptional sequences of invertible sheaves on rational surfaces. We construct full strongly exceptional sequences for a large class of rational surfaces. For the case of toric surfaces we give a complete classification of full strongly exceptional sequences of invertible sheaves.

A counterexample to King's conjecture

Kings conjecture states that on every smooth complete toric variety

RESOLUTIONS AND COHOMOLOGIES OF TORIC SHEAVES: THE AFFINE CASE

X

TILTING BUNDLES ON RATIONAL SURFACES AND QUASI-HEREDITARY ALGEBRAS

there exists a strongly exceptional collection which generates the bounded derived category of

Graded rings and equivariant sheaves on toric varieties

X

Examples for exceptional sequences of invertible sheaves on rational surfaces

and which consists of line bundles. We give a counterexample to this conjecture. This example is just the Hirzebruch surface

Moduli for Equivariant Vector Bundles of Rank Two on Smooth Toric Surfaces

\mathbb{F}_2

Divisorial cohomology vanishing on toric varieties.

iteratively blown up three times, and we show by explicit computation of cohomology vanishing that there exist no strongly exceptional sequences of length 7.

Resolutions and Moduli for Equivariant Sheaves over Toric Varieties

We study equivariant resolutions and local cohomologies of toric sheaves for affine toric varieties, where our focus is on the construction of new examples of indecomposable maximal Cohen–Macaulay modules of higher rank. A result of Klyachko states that the category of reflexive toric sheaves is equivalent to the category of vector spaces together with a certain family of filtrations. Within this setting, we develop machinery which facilitates the construction of minimal free resolutions for the smooth case as well as resolutions which are acyclic with respect to local cohomology functors for the general case. We give two main applications. First, over the polynomial ring, we determine in explicit combinatorial terms the ℤn-graded Betti numbers and local cohomology of reflexive modules whose associated filtrations form a hyperplane arrangement. Second, for the nonsmooth, simplicial case in dimension d ≥ 3, we construct new examples of indecomposable maximal Cohen–Macaulay modules of rank d – 1.