HELICES ON SOME FANO THREEFOLDS - CONSTRUCTIVITY OF SEMIORTHOGONAL BASES OF K-O

The paper deals with the properties of helices on the level of the Grothendieck group K-O(X). For Fano threefolds with Pic X congruent to Z (the simplest from the point of view of helix theory) there are considered the semiorthogonal bases of K-O(X), viewed as a Z-module, which arise as a natural generalization of images in K-O of foundations of helices. For these threefolds equations are derived, which play the role analogous to that of Markov equation for helices on P-2. With the help of these equations the semiorthogonal bases of K-O are classified module action of mutations, i.e. the ''K-O-constructivity'' of helices is proved.