VAN DER VRIES. — MULTIPLE POINTS OP TWISTED CURVES. 511 



mfi— (m — 2)k+l ^ ^ (/x + 2) - P. 



If there are still points left at our disposal, we may use them in different 

 ways, and may cause the residual curve to break up in different ways. 

 The complete intersection of Kn and Sfj, is of order 2^; and as Cm is of 

 order m the residual is of order 2 fi — m. As the comjjlete intersection 

 has a point of multiplicity 2 k at A, the residual has a point of multi- 

 plicity 2 /(,• — m + 2 there. Every edge of A'o meets Sfj, in ^' + 1 points 

 of which k are at A. If we can take jw — k additional points on some of 

 these edges, these edges will contain fx -{- 1 points in all and will lie com- 

 pletely on Sfj,. Therefore if there are enough points left at our disposal 

 to enable us to take fi — k additional points on each of 2 jtt — w — 1 

 edges of K2, the residual will consist of these 2jw — m — 1 edges of iTg 

 and (since the entire residual is of order 2 ft — m) another straight line, 

 which can only be another edge of K^. The residual can then be made 

 to consist entirely of straight lines if 



(fi — k) {2n — m — l) ^(x {^+ 2) - P - mfi + (m — 2) k — 1, 



i. e. if (fi-k-1) (fi-k-2)'^l', 



which can only occur if^ = jM — 1, or k = [x — 2. We may then always 

 take k ^^ i^ — 1, that is take St^ to be a monoid. Thus ^ = ^ + 1 = 



— or — - — according as m is even or odd. Therefore : — 

 2 2 * 



A unicursal curve of order m that has a point of multiplicity m — 2 can 

 always be considered as the partial intersection of a quadric cone and a 



monoid of order jw, where ^ z= — or — - — according as m is even or odd. 



Li U 



Thus the quintic curve with a triple point can be obtained, as we have 

 seen before, as the partial intersection of a quadric cone and a cubic 

 monoid that have the same vertex. The cubic monoid is determined by 

 15 points, but 4 of these must be taken off the quadric in order not to 

 have the monoid contain the quadric as a component. We thus have II 

 points at our disposal. In order to make the cubic monoid contain a 

 quintic curve that has a triple point at the vertex of the monoid, we 

 must make it contain 3'5 — 2*3-f-l or 10 additional points of the 

 curve. We can do this and still have one point left at our disj)osal. This 

 point can be taken on any generator of the cone, causing this generator 

 to be the residual intersection of the cone and the monoid. 



Similarly, a sextic curve with a quadruple point can be obtained as the 



