WITH ORDINARY SINGULARITIES IN A PROJECTIVE SPACE.

In our previous paper [5] we have proved a theorem of completeness of characteristic systems for analytic families of surfaces with ordinary singularities in ambient threefolds. In this paper we examine the application of the theorem to surfaces with ordinary singularities in a projective 3-space. The theorem asserts that the characteristic systems of a maximal analytic family of surfaces with ordinary singularities in an ambient threefold are complete if the surfaces are semi-regular. We show, in Section 2, that, in a projective 3-space, the semi-regularity coincides with the regularity and that the regularity is equivalent to the linear independence of certain simultaneous linear equations (Theorem 2). We reformulate the theorem of completeness of characteristic systems for analytic families of surfaces with ordinary singularities in a projective 3-space (Theorem 1). In Section 4 we prove some criteria of regularity (Theorems 5, 6 and 7). The problem of proving the completeness of the characteristic systems of complete continuous systems of surfaces with ordinary singularities in a projective 3-space has been proposed by 0. Zariski (see Zariski [9], p. 99). Our results in Sections 2 and 4 may be considered as partial answers to this problem. In Section 5 we consider several concrete examples of surfaces. First we show with the aid of the criteria that certain classical examples of surfaces are regular. We theii examine surfaces whose double curves are non-singular complete intersections and find that they are not regular except some special cases whereas they form maximal families with complete characteristic systems. Apparently this result indicates that the requirement of semi-regularity imposes a strong restriction on surfaces with ordinary singularities. Finally we examine the application of Theorem 1 to canonical surfaces of irregularity 0, of genus 4 and of order 7 studied earlier by Enriques [2] and by Maxwell [7]. 1. Preliminaries. We denote by W a projective 3-space defined over the field C of complex numbers. We denote a point in W by 1v = (wo, w1, wu2, w3),