Exceptional Collections, Mutations and Helixes

Summary

We shall give a general axiomatic presentation of the theory of helices and introduce some general definitions and notations.

Research on exceptional bundles was started in Moscow University after a lecture given by A.N. Tyurin given in the autumn of 1984 on a preprint of [1], In that paper a theorem is given describing the possible Chern classes which a stable bundle on P² can have. Exceptional bundles appeared as some sort of boundary points. The results of the first one and a half years of our work were presented in [4]. Papers [3] and [6] together with subsequent papers represent the research of the following one and a half years. Most of the papers use a technique which should be called Helix Theory.

The definition of a helix and the first results about helices appeared in [4]. The key lemma 2.2 of that paper and the first version of the definition of a helix are due to Gorodentsev [3]. These constructions were a generalisation to arbitrary dimensions of a method of Rudakov which assigned an exceptional bundle on a projective plane to a pair of exact sequences [5]. The word “helix” and the idea of considering a helix as an infinite system of bundles with some form of periodicity is due to W.N. Danilov.

Further development of the notion of a helix was connected with applications of the primary ideas in new contexts. This was done by Gorodentsev for arbitrary categories of coherent sheaves [3], by Rudakov for a category of symmetric sheaves on a two-dimensional quadric [6] and by others in subsequent papers.