Helix Theory and Nonsymmetrical Bilinear Forms

In this paper we will discuss the connections between the description of the combinatorical structure of the set of exceptional vector bundles and some natural questions about the unimodular (nonsymmetrical) bilinear forms. For investigation of the set of exceptional vector bundles (such bundles that Exti (E, E = 0 for i ≠ 0 and dim Ext0 (E, E = 1) there was developed five years ago in [GR], [G1], [G2] the theory of helices. In section 1 we give a review of the main results of this theory. In section 2 we formulate some questions about classes of exceptional vector bundles in Grothendieck group K 0 in terms of a lattice with nonsymmetrical nondegenerate integer bilinear form. Arithmetical problems of helix theory are equivalent to the description of the orbits of isometries and orbits of the natural action of braid group on semiorthogonal bases of a lattice. In section 3 we speak about the general properties of nonsymmetrical forms and their isometries and then give some examples in section 4.