Geodesic
Isometries of semi-orthogonal forms on a $\mathbb Z$-module of rank 3
Izvestiya. Mathematics, Tome 77 (2013) no. 1, pp. 44-86 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study the isometry groups of semi-orthogonal forms (that is, forms whose Gram matrix in some basis is upper triangular with ones on the diagonal) on a $\mathbb Z$-module of rank 3. Such forms have a discrete parameter: the height (the trace of the dualizing operator + 3). We prove that the isometry group is either $\mathbb Z$ or $\mathbb Z_2\times\mathbb Z$, list all the cases when it is a direct product and describe the generator of order 2 in that case. We also describe a generator of infinite order for many particular values of the height.
Keywords: quadratic forms on modules over rings. 
@article{IM2_2013_77_1_a4,
     author = {S. A. Kuleshov},
     title = {Isometries of semi-orthogonal forms on a~$\mathbb Z$-module of rank~3},
     journal = {Izvestiya. Mathematics},
     pages = {44--86},
     year = {2013},
     volume = {77},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2013_77_1_a4/}
}
TY  - JOUR
AU  - S. A. Kuleshov
TI  - Isometries of semi-orthogonal forms on a $\mathbb Z$-module of rank 3
JO  - Izvestiya. Mathematics
PY  - 2013
SP  - 44
EP  - 86
VL  - 77
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/IM2_2013_77_1_a4/
LA  - en
ID  - IM2_2013_77_1_a4
ER  - 
%0 Journal Article
%A S. A. Kuleshov
%T Isometries of semi-orthogonal forms on a $\mathbb Z$-module of rank 3
%J Izvestiya. Mathematics
%D 2013
%P 44-86
%V 77
%N 1
%U http://geodesic.mathdoc.fr/item/IM2_2013_77_1_a4/
%G en
%F IM2_2013_77_1_a4
S. A. Kuleshov. Isometries of semi-orthogonal forms on a $\mathbb Z$-module of rank 3. Izvestiya. Mathematics, Tome 77 (2013) no. 1, pp. 44-86. http://geodesic.mathdoc.fr/item/IM2_2013_77_1_a4/

[1] A. L. Gorodentsev, “Transformations of exceptional bundles on $\mathbf P^n$”, Math. USSR-Izv., 32:1 (1989), 1–13 | DOI | MR | Zbl

[2] S. A. Kuleshov, “On moduli spaces for stable bundles on quadrics”, Math. Notes, 62:6 (1997), 707–725 | DOI | DOI | MR | Zbl

[3] S. A. Kuleshov, D. O. Orlov, “Exceptional sheaves on del Pezzo surfaces”, Russian Acad. Sci. Izv. Math., 44:3 (1995), 479–513 | DOI | MR | Zbl

[4] A. N. Rudakov, “The Markov numbers and exceptional bundles on $\mathbf P^2$”, Math. USSR-Izv., 32:1 (1989), 99–112 | DOI | MR | Zbl

[5] J. M. Drezet, J. Le Potier, “Fibrés stables et fibrés exceptionnels sur $\mathbb P^2$”, Ann. Sci. École Norm. Sup. (4), 18:2 (1985), 193–243 | MR | Zbl

[6] D. Yu. Nogin, “Spirals of period four and equations of Markov type”, Math. USSR-Izv., 37:1 (1991), 209–226 | DOI | MR | Zbl

[7] S. A. Kuleshov, A. N. Rudakov, Tseloznachnye bilineinye formy, Tr. VI Kolmogorovskikh chtenii, YaGPU, Yaroslavl, 2008

[8] A. I. Borevich, I. R. Shafarevich, Number theory, Academic Press, New York–London, 1966 | MR | MR | Zbl | Zbl