Seminarium Nieliniowość i Geometria

Desargues maps provide simple incidence-geometricinterpretation to (non-commutative in general) discrete KP system. Thedefining condition is collinearity of images of vertices of basicsimplices of the A-type root lattice. In trying to find thecorresponding interpretation of reductions to Painleve' equations (asan example we present detailed analysis of reduction to q-P_III) one isforced to consider Desargues maps into a projective line where,however, the defining condition loses its meaning. We propose asolution of this problem in terms of the so-called quadrangular set ofpoints, which plays a prominent role in construction of themultiplication in a division ring underlying the projective line. Thestructure of the division ring may provide geometric meaning forfurther reductions of Desargues maps. As an example, we show how thecircle-geometry interpretation of the discrete KP equation, consideredby Konopelchenko and Schief, fits into the Desargues maps scheme.