Exceptional vector bundles on projective spaces. (English) Zbl 0646.14014

Gorodentsev, A. L.; Rudakov, A. N.

Duke Math. J. 54, 115-130 (1987).

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

References:

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