On invariants of point correspondences between three curves of projective planes

Background. We continue studying the geometry of point correspondences between manifolds. The main problem of the paper is to find invariants of point correspondences under consideration according to the transformation group, which is a direct product of projective transformation groups of planes of curves and to find geometric meanings of invariants. Materials and methods. In the way of investigation we use the exterior differentiation, tensor analysis and G. F. Laptev invariant methods. Rezults. It is proved, that invariant moving frames of all three curves of correspondences are determined by fours-order differential neighbourhood. We determine three invariant of the second-order neighbourhood, four invariants of the third-order neighbourhood and six invariants of the fours-order neighbourhood and find some geometric сharacteristics of invariants. The connection among studied correspondences and 3-webs is schown. Some partial cases are considered. Conclusions. Invariant methods of investigation of point correspondences between manifolds help to find very useful properties, which are not depend on the choice of frames. Connections between theories of correspondences in question and 3-webs help to develop both geometries.