Электронный каталог

Статья

Dulatov, I.T.
On Quadratic Transformations of the Fano Plane / I.T.Dulatov, M.D.Malykh, A.L.Sevastianov, A.V.Zorin. – Text : electronic // Discrete and Continuous Models and Applied Computational Science. – 2026. – Vol. 34, No. 2. – P. 201-213. – URL: https://doi.org/10.22363/2658-4670-2026-34-2-201-213. – Bibliogr.: 20.

The importance of studying quadratic Cremona transformations over algebraically non-closed fieldsfor the theory of Kahan difference schemes for dynamical systems with a quadratic right-hand side is discussed.Cremona transformations of the projective plane over a Galois field of size 2, i.e., the Fano plane, are considered.The notation proposed by J. Rosanes is used to describe quadratic Cremona transformations. The relationshipbetween quadratic transformations and matrix pencils is described. Quadratic transformations without singularpoints are called regular. The Sage system is used to implement procedures that convert a quadratic transforma-tion into a permutation of 7 points of the Fano plane (an element of the symmetric group𝑆7) and a permutationof 7 points of the plane into a quadratic transformation. By enumerating all quadratic transformations, it isproved that regular quadratic transformations generate the entire permutation group of the Fano plane. Thistheorem is analogous to Noether’s theorem over an algebraically non-closed field. It is proved that regular qua-dratic Cremona transformations have even order, and a description of the corresponding permutations of thegroup𝑆7is given. It is shown that a permutation always corresponds to some quadratic Cremona transformation,but this transformation is not always regular. The resulting quadratic transformations can be supplemented bya rule that resolves ambiguities at fundamental points. An example of a quadratic transformation for whichsuch resolution is impossible is given. This leads to a natural classification of quadratic transformations of theFano plane



Спец.(статьи,препринты) = С 138 - Геометрия. Риманова геометрия. Геометрия Лобачевского
ОИЯИ = ОИЯИ (JINR)2026