Vol. 54, No. 1 DUKE MATHEMATICAL JOURNAL (C) 1987 EXCEPTIONAL VECTOR BUNDLES ON PROJECTIVE SPACES A. L. GORODENTSEV AND A. N. RUDAKOV Introduction 115 1. Exceptional collections and helixes 116 2. Mutations of helixes 119 3. Spectral sequences associated with a helix 121 4. Exceptional bundles on P 2 125 5. Helixes on P 2 and the Markov equation 126 References 130 In this article we construct an infinite set of exceptional vector bundles on P n. We prove that all exceptional bundles on P are constructed in this way and their dimensions are Markov numbers. As a by-product we get some spectral sequences that are generalizations of the Beilinson spectral sequences. In the paper we shall study algebraic coherent sheaves on p n. We shall call such a sheaf F an exceptional sheaf iff dim Ext 0 (F, F) 1, Exti(F,F) =0 wherei>0. Our ground field is a complex number field but we believe that almost all our results are valid over an arbitrary field. An exceptional sheaf F on P must be locally free and the associated vector bundle we shall also call exceptional. For n 2 such bundles were studied in [2] but there they had a different definition, an exceptional sheaf means a stable sheaf with discriminant less then 1/2. We prove that for P this definition and ours are equivalent. It is an interesting question whether an exceptional sheaf on P for n > 2 is stable. The first examples of exceptional sheaves are (.0(i). It turns out that other exceptional sheaves on are naturally composed in infinite series (Ei), 7/ and we call them helixes. The series (O(i)) is an example of helix. We define mutations of helixes and via these mutations we obtain an infinite set of helixes on P ". We prove that for P 2 this set consists of all helixes. Each helix produces two spectral sequences in the manner of the Beilinson spectral sequences for the helix (O(i)). We find also that for each helix on p2 there is a corresponding integral solution of the Markov equation x 2 + y2 + z 2 3xyz. It is known that integral Received September 29, 1986 115