# zbMATH — the first resource for mathematics

Exceptional vector bundles on projective spaces. (English) Zbl 0646.14014
A coherent algebraic sheaf $$F$$ on $$\mathbb{P}^ n = \mathbb{P}^ n(\mathbb{C})$$ is called exceptional if $$\dim(Ext^ 0(F,F)) = 1$$ and the higher $$Ext^ i(F,F)$$ vanish. When $$n = 2$$, this definition agrees with that of J. M. Drezet and J. Le Potier [Ann. Sci. Ec. Norm. Super., IV. Ser. 18, 193-243 (1985; Zbl 0586.14007)] : stable, with discriminant $$< 1/2$$. For example, each $${\mathcal O}(i)$$ is exceptional. The authors extend this list by constructing an infinite collection $$(E_ i)$$ of exceptional sheaves, called helixes. Each helix corresponds to an integral solution of the Diophantine equation $$x^ 2 + y^ 2 + z^ 2 = 3xyz$$, and gives rise to two spectral sequences, generalizing Beilinson’s for $${\mathcal O}(i)$$.
Reviewer: R.Speiser

##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
##### Keywords:
exceptional algebraic sheaf; helix
Full Text:
##### References:
 [1] J. W. S. Cassels, An introduction to Diophantine approximation , Hafner Publishing Co., New York, 1972. · Zbl 0077.04801 [2] J.-M. Drezet and J. Le Potier, Fibrés stables et fibrés exceptionnels sur $$\mathbf P_ 2$$ , Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 193-243. · Zbl 0586.14007 · numdam:ASENS_1985_4_18_2_193_0 · eudml:82158 [3] J.-M. Dreset, Fibrés exceptionnels et suite spectrale de Beilinson généralisée sur $$\mathbbP_2(\mathbbC)$$ , Math. Ann. (1986), B.275, h.1. · Zbl 0578.14013 · doi:10.1007/BF01458581 · eudml:164131 [4] S. Mukai, On the moduli space of bundles on $$K3$$ surfaces I , · Zbl 0674.14023 [5] Ch. Okonek, M. Schneider, and H. Spindler, Vector bundles on complex projective spaces , Progress in Mathematics, vol. 3, Birkhäuser Boston, Mass., 1980. · Zbl 0438.32016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.