Spirals of period~4 and equations of Markov type

D. Yu. Nogin

Izvestiya. Mathematics , Tome 37 (1991) no. 1, pp. 209-226

Résumé

Equations which connect the rank of the elements of the coil of a spiral of period four and certain parameters of the exceptional pairs in the coil are derived. These equations are analogues of Markov's equation, which is satisfied by the ranks of the exceptional bundles which form the coil of a spiral on $\mathbf P^2$. These equations are used to prove the constructivity of spirals on such manifolds as $\mathbf P^3$ and rational ruled surfaces. Constructivity means that by surgery one can bring the spiral to a canonical form, up to equivalence in the Grothendieck group $\operatorname{K_0}$. The application to the theory of exceptional bundles of the methods of this paper are illustrated at the level of $\operatorname{K_0}$, via a proof of geometric constructivity of spirals on the ruled surface $\mathbf F_1$ – every spiral can be reduced to a canonical spiral of one-dimensional bundles by means of surgery.