By a geometrical process, depending on the simplest Cremona transformation (see below), it is possible not only to enumerate the multiple points and lines contained in any singularity, but also to construct a penultimate form to indicate the arrangement of these components. (Scott, J. Math.,' v. 14, pp. 301-325; v. 15, pp. 221-243; 1892-93).
The various processes lead to the following conclusions as to the content of a complex sin gularity. The point-equation by its lowest terms gives the order, k, of the singularity; the line-equation gives the class, I. If the singu larity has all its tangents coincident (if it has not, it can be broken up into simpler ones) then order + class -.number of intersections with the tangent at the point, and thus the class is known without reference to the line-equation. Coincident tangents may lead to other multiple points, or to chains of multiple points, con tained in the singularity; if the total number of dps in these is h, the singularity has h latent double points. It has also latent double lines, h' in number, and it has been proved that h. The singularity of order k, class I, ex cess h (supposed irreducible), involves ik(k — 1) + h dps, of which k —I are cusps, and WI.-- 1) + h dls, of which I— I are inflexional tangents. The singularity is equivalent to those components as regards Pliicker's equations and as regards the genus; hut not in the number of conditions imposed on the curve.
It is obvious that the multiple points of a curve cannot always be chosen arbitrarily. A sextic, e.g., can have 10 tips; but these, chosen arbitrarily, would impose 30 conditions, whereas 27 determine the sextic. The 27 conditions cannot be imposed by 9 dps arbitrarily chosen, for these would determine only the reducible sextic where v is the cubic through the 9. If the 9 admit of a proper sextic, u 0, they allow one degree of freedom, for all curves satisfy the conditions. Hence the 9 are not arbitrary. The theorems of geometry on a curve, and linear systems of curves, lead to general results of this character, but there are evidently many special theorems as to the posi tion of singularities to be formulated for curves of specified order or class.
By means of a birational transformation, either of the whole plane or of the one curve, it is possible to change any curve into one, in general of different order, with no point-singu larities except simple nodes. Let P, (x, y,
become P', (x', y', it) (represented on a second for distinctness), where x', y', a' are given y, (x, y, a) :#1 y, a); then corresponding to any point P of the first plane // there is one point P' of the second plane II'; but the converse does not hold. The straight lines of II', Ix'±my'+nz'=0, corre spond to the co transformation-curves in II, Ms+ m44+ nO,=-,0, of order a; to a single point P' of II', given as the intersection of two lines, there correspond in II all the K variable inter sections of two 's, Pa; the corre spondence of the two planes is said to be lK-to-one,s (K, 1). Since the O's may have fixed points, simple or multiple, th, of orders pi, pi, . ., a may be less than al.
If i.e., if one point of II corresponds to one point of II', the equations of transformation are reversible; not only is II' expressed ration in terms of II by these equations, but also II is expressed rationally in terms of IF by x:y:z--=.11)1 (x', y', (a', y, (x', y', z'), the reverted equations, which are of the same order a. The transformation is a birational transformation of the plane, usually called a Cremona transformation. In the simplest Cre mona transformation the O's are conics through three Axed points; the equations of transforma tion are therefore of the second degree; this is a reversible quadratic transformation. An im portant theorem is that every Cremona trans formation can be accomplished by a succession of these reversible quadratic transformations.
If K> 1, the equations cannot be reverted; the transformation is not birational for the whole plane. But ?f Pi trace a curve F, P' traces the corresponding curve F'; the points trace some curve, f (different from F), the com panion curve of F. If we ignore all of II ex cept F, that is, if we confine ourselves to the two curves F. F', the correspondence becomes one-to-one; with the help of the equation F==. 0 the equations of transformation can be reverted, and the transformation becomes birational. Such a birational transformation of a curve is called a Riemann transformation. Thus a Riemann transformation is birational for the one curve only, while a Cremona transforma tion is birational for the whole plane, and there fore for every curve in the plane.