On isolated singularities of surfaces which do not affect the conditions of adjunction (Part II.)

Abstract

In the first part of this paper I investigated the nature of the isolated singular points which can appear on an algebraic surface without affecting the conditions of adjunction and the arithmetic genus of the surface. It appeared that such points have neighbourhoods analysable into connected chains or trees of curves, each rational and of virtual grade − 2, and arranged either in a single chain (giving a binode, or a conic node if there is only one curve) or in three chains of, say, n, p, q curves respectively, one end curve of each chain meeting one which we may call the central curve of the tree. The values of n, p, q are not arbitrary, but satisfy either p = q = 1, or n ≤ 4, p = 2, q = 1; the unodes given by the former case we call U, those given by the latter case U, with a suffix indicating the reduction in class in each case. I pointed out the similarity between these results and Coxeter's enumeration of groups generated by reflexions, there being a one-one correspondence between these singularities and Coxeter's groups, with the restriction that the only groups with which we are concerned are those in which any two primes of symmetry are inclined at either π/2 or π/3. In fact, the curves which make up a complete neighbourhood correspond to the bounding primes of a fundamental region, two curves which do not meet corresponding to mutually perpendicular, primes, and two which do to primes inclined at π/3.