Abstract
We show that the non-commutative widths for flopping curves on smooth three-folds introduced by Donovan–Wemyss are described by Katz’s genus zero Gopakumar–Vafa invariants.
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References
Bryan J., Katz S., Leung N.C.: Multiple covers and integrality conjecture for rational curves on Calabi-Yau threefolds. J. Algebraic Geom. 10, 549–568 (2001)
Bridgeland T.: Flops and derived categories. Invent. Math 147, 613–632 (2002)
Chen J.-C.: Flops and equivalences of derived categories for three-folds with only Gorenstein singularities. J. Differential. Geom 61, 227–261 (2002)
Donovan, W., Wemyss, M.: Noncommutative deformations and flops, preprint, arXiv:1309.0698
Huybrechts D.: Fourier–Mukai Transforms in Algebraic Geometry, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2006)
Katz S.: Genus zero Gopakumar–Vafa invariants of contractible curves. J. Differ. Geom. 79, 185–195 (2008)
Katz S., Morrison D.R.: Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups. J. Algebraic Geom. 1, 449–530 (1992)
Orlov D.: On equivalences of derived categories and K3 surfaces. J. Math. Sci 84, 1361–1381 (1997)
Reid, M.: Minimal models of canonical three-folds. In: Iitaka S. (ed.) Algebraic Varieties and Analytic Varieties. Adv. Stud. Pure Math, Kinokuniya, Tokyo, and North-Holland, Amsterdam, vol. 1, pp. 131–180
Seidel P., Thomas R.P.: Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108, 37–107 (2001)
Toda Y.: On a certain generalization of spherical twists. Bull. de la SMF 135, 97–112 (2007)
Toda, Y.: Stability conditions and crepant small resolutions. Trans. Amer. Math. Soc., 360, 6149–6178, arXiv:math/0512648 (2008)
Van den Bergh M.: Three dimensional flops and noncommutative rings. Duke Math. J. 122, 423–455 (2004)
