Формальное представление условий инцидентности в многомерных проективных пространствах

Abstract

Synthetic and analytical methods are usually used to investigate multidimensional spaces and sets of subspaces. Shortcomings of a synthetic method — the appeal to spatial imagination and the researcher´s intuition, impossibility of formalization, need of creation of big and complex logical constructions don´t allow to go beyond four-dimensional space, with rare exceptions. The theory of enumerative geometry submitted as geometry of conditions with a basic element – a multidimensional flags incidence condition allows solve many problems which up to now were considered as insoluble ones. The simplest of such problems is a classical problem for calculation of final number of given space’s subspaces meeting the set of given conditions (the normal problem of algebraic geometry). A more serious problem – calculation a number and values of algebraic characteristics for given variety in given space. For this problem solution it was necessary to develop a formalized method, as well as technique algorithmization. This problem has been solved by professor. V.Ya. Volkov in his doctoral dissertation by means of developed by him so-called e-calculations. For an understanding of e-calculation fundamentals or calculation of Schubert conditions a rather good mathematical background and method promotion are necessary. The last demands consideration of various approaches to conditions calculation problem. In the present paper are considered the simplest cases of a tabular method for conditions calculation, in relation to conditions of incidence which is understood in a general sense. Calculation of one-, two- and ((k + 1) (n – k) – 1)-dimensional conditions are considered. Conditions calculation formalization and incidence conditions reduction are explained. The problem about a final number of straight lines crossing the set number of k-planes in n-dimensional space is considered as an example. In particular, the problem about a number of straight lines crossing some number of the set straight lines can be correct only in three- and four-dimensional spaces. Conditions of the minimum multiplicity equal to three exist only in (3k + 1)-dimensional spaces. Conditions of the multiplicity equal to four exist only in odd dimensionality spaces. And so on. The concrete number of straight lines in all cases can be counted by reduction of the corresponding conditions.