On a property of a degenerate diametral surface of an algebraic surface of the spaceE m

Abstract

We study the structure of the degenerate diametral surface Dk(~) of arbitrary order k, 1 < k <_ n -1, of an algebraic surface of order n. The case k = 4 is considered in detail.Suppose that in real m-dimensional Euclidean space E m an (m -1):dimensional algebraic surface F,~ of order n is defined by the equation geometric interpretation of the degenerate diametral surfaces Dk(g), k > t, conjugate to the vector E. A geometric interpretation of a degenerate diametral quadric is given in [3].We consider the structure of the diametral surface Dk(E) of arbitrary order k, i < k < n -1, in the case when its degeneracy is due to the leading forms ~o,~-t(~) (l = 0, k) having a common factor X(Z), a form of degree p > 1 such thatx(E ) = 0 for the chosen direction E. We have proved the following theorem.