Hostname: page-component-7479d7b7d-k7p5g Total loading time: 0 Render date: 2024-07-14T01:41:23.089Z Has data issue: false hasContentIssue false
  • English
  • Français

Differential Graded Quivers of Smooth Rational Surfaces

Part of: Special varieties Surfaces and higher-dimensional varieties (Co)homology theory

Published online by Cambridge University Press:  15 December 2016

Agnieszka Bodzenta
Agnieszka Bodzenta*
Affiliation:
School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK (a.bodzenta@ed.ac.uk)
Get access Rights & Permissions [Opens in a new window]

Abstract

Let X be a smooth rational surface. We calculate a differential graded (DG) quiver of a full exceptional collection of line bundles on X obtained by an augmentation from a strong exceptional collection on the minimal model of X. In particular, we calculate canonical DG algebras of smooth toric surfaces.

Keywords

derived categoriesexceptional collectionsDG categorytoric varieties

MSC classification

Secondary: 14F05: Sheaves, derived categories of sheaves and related constructions 14J26: Rational and ruled surfaces 14M25: Toric varieties, Newton polyhedra
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 4 , November 2017 , pp. 859 - 876
Copyright
Copyright © Edinburgh Mathematical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Bodzenta, A., DG categories and exceptional collections, Proc. Am. Math. Soc. 143(5) (2015), 1909–1923.Google Scholar
2. Bondal, A. I., Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat. 53(1) (1989), 25–44.Google Scholar
3. Bondal, A. I. and Kapranov, M. M., Framed triangulated categories, Mat. Sb. 181(5) (1990), 669–683.Google Scholar
4. Bridgeland, T., T-structures on some local Calabi–Yau varieties, J. Alg. 289(2) (2006), 453–483.Google Scholar
5. Fulton, W., Introduction to toric varieties, Annals of Mathematics Studies, Volume 131 (Princeton University Press, 1993).Google Scholar
6. Hille, L. and Perling, M., Exceptional sequences of invertible sheaves on rational surfaces, Compositio Math. 147(4) (2011), 1230–1280.Google Scholar
7. Hille, L. and Perling, M., Tilting bundles on rational surfaces and quasi-hereditary algebras, Annales Inst. Fourier 64(2) (2014), 625–644.Google Scholar
8. Orlov, D. O., Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat. 56(4) (1992), 852–862.Google Scholar
9. Seidel, T., Homological mirror symmetry for the quartic surface, Preprint (arXiv:math/0310414 [math.SG]; 2003).Google Scholar