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The Markov numbers and exceptional bundles on \(P^ 2\). (English. Russian original) Zbl 0661.14017

Math. USSR, Izv. 32, No. 1, 99-112 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 1, 100-112 (1988).
The author considers exceptional vector bundles E on \({\mathbb{P}}^ 2({\mathbb{C}})\), i.e. coherent sheaves E on \({\mathbb{P}}^ 2\) such that \(Hom(E,E)={\mathbb{C}}\), \(Ext^ 1(E,E)=0\), \(Ext^ 2(E,E)=0\), and the Markov diophantine equations \(x^ 2+y^ 2+z^ 2=3xyz\) (introduced by A. A. Markov in 1880). He defines triples of neighboring exceptional vector bundles and establishes a correspondence between these triples and the solutions of the Markov equation. It is proved that the dimensions of the bundles that constitute a neighboring triple are solutions of the Markov equations.
In the paper there is also described a construction of the exceptional vector bundles on \({\mathbb{P}}^ 2\). The paper contains an answer to a problem of A. N. Tjurin about the representation of the exceptional vector bundles and establishes many auxiliary results of homological algebra.
Reviewer: D.Ştefănescu

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
11D25 Cubic and quartic Diophantine equations
14B15 Local cohomology and algebraic geometry
18G35 Chain complexes (category-theoretic aspects), dg categories
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