Geodesic
Spirals of period 4 and equations of Markov type
Izvestiya. Mathematics, Tome 37 (1991) no. 1, pp. 209-226 
D. Yu. Nogin. Spirals of period 4 and equations of Markov type. Izvestiya. Mathematics, Tome 37 (1991) no. 1, pp. 209-226. http://geodesic.mathdoc.fr/item/IM2_1991_37_1_a8/
@article{IM2_1991_37_1_a8,
     author = {D. Yu. Nogin},
     title = {Spirals of period~4 and equations of {Markov} type},
     journal = {Izvestiya. Mathematics},
     pages = {209--226},
     year = {1991},
     volume = {37},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1991_37_1_a8/}
}
TY  - JOUR
AU  - D. Yu. Nogin
TI  - Spirals of period 4 and equations of Markov type
JO  - Izvestiya. Mathematics
PY  - 1991
SP  - 209
EP  - 226
VL  - 37
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/IM2_1991_37_1_a8/
LA  - en
ID  - IM2_1991_37_1_a8
ER  - 
%0 Journal Article
%A D. Yu. Nogin
%T Spirals of period 4 and equations of Markov type
%J Izvestiya. Mathematics
%D 1991
%P 209-226
%V 37
%N 1
%U http://geodesic.mathdoc.fr/item/IM2_1991_37_1_a8/
%G en
%F IM2_1991_37_1_a8

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Beilinson A. A., “Kogerentnye puchki na $\mathbf{P}^n$ i problemy lineinoi algebry”, Funkts. analiz i ego prilozh., 12:3 (1978), 68–69 | MR | Zbl

[2] Gorodentsev A. L., “Perestroiki isklyuchitelnykh rassloenii na $\mathbf{P}^n$”, Izv. AN SSSR. Ser. matem., 52:1 (1988), 3–15 | MR

[3] Gorodentsev A. L., “Isklyuchitelnye rassloeniya na poverkhnostyakh s podvizhnym antikanonicheskim klassom”, Izv. AN SSSR. Ser. matem., 52:4 (1988), 740–750

[4] Gorodentsev A. L., Rudakov A. N., “Exceptional vector bundles on the protective spaces”, Duke Math. J., 64:1 (1987), 115–130 | DOI | MR

[5] Kapranov M. M., “Proizvodnaya kategoriya kogerentnykh puchkov na mnogoobraziyakh Grassmana”, Izv. AN SSSR. Ser. matem., 48:1 (1984), 192–202 | MR

[6] Kapranov M. M., “Proizvodnaya kategoriya kogerentnykh puchkov na kvadrike”, Funkts. analiz i ego prilozh., 20:2 (1986), 67 | MR | Zbl

[7] Kvichanskii A. V., Nogin D. Yu., Isklyuchitelnye nabory na lineichatykh poverkhnostyakh. Algebraicheskaya geometriya, Dep. v VINITI ot 9.08.1988, No 6418-V-88

[8] Rudakov A. N., “Isklyuchitelnye rassloeniya na $\mathbf{P}^2$ i chisla Markova”, Izv. AN SSSR. Ser. matem., 52:1 (1988), 100–112 | MR

[9] Rudakov A. N., “Isklyuchitelnye rassloeniya na kvadrike”, Izv. AN SSSR. Ser. matem., 52:4 (1988), 782–812