Exceptional Sequences of Invertible Sheaves on Rational Surfaces

Abstract

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.

40 pages, 5 figures, requires packages ams*, enumerate, xy, geometry, minor changes

References in corpus (6)

• Exceptional collections for Grassmannians of isotropic lines
• Brane Tilings and Exceptional Collections
• Exceptional collections and D-branes probing toric singularities
• Moduli spaces for Bondal quivers
• Projective toric varieties as fine moduli spaces of quiver representations
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• Exceptional Sequences on Rational C*-Surfaces
• Exceptional sequences and spherical modules for the Auslander algebra of $k[x]/(x^t)$
• Frobenius-Perron Theory of Representations of Quivers
• Exceptional Sequences of Line Bundles and Spherical Twists - a Toric Example
• Embeddings of algebras in derived categories of surfaces
• Exceptional collections in surface-like categories
• Homological mirror symmetry of toric Fano surfaces via Morse homotopy
• SYZ mirror of Hirzebruch surface ${\mathbb{F}}_k$ and Morse homotopy
• Classification of full exceptional collections on smooth toric Fano varieties with Picard rank two
• Mutations of Numerically Exceptional Collections on Surfaces