LIBRARY University of California. RECEIVED BY EXCHANGE Class Cbe TRmx>ersit£ ot Gbtcago FOUNDED BY JOHN D. ROCKEFELLER ON THE RESOLUTION OF HIGHER SINGULARITIES OF ALGEBRAIC CURVES INTO ORDINARY NODES BY B. M. WALKER A DISSERTATION Submitted to the Faculties of the Graduate Schools op Arts, Literature, and Science in Candidacy for the Degree of Doctor of Philosophy DEPARTMENT OF MATHEMATICS CHICAGO 1906 V Zbc TElntvexsity of Cbtcaao FOUNDED BY JOHN D. ROCKEFELLER ON THE RESOLUTION OF HIGHER SINGULARITIES OF ALGEBRAIC CURVES INTO ORDINARY NODES BY B. M. WALKER A DISSERTATION Submitted to the Faculties of the Graduate Schools of Arts, Literature, and Science in Candidacy for the Degree of Doctor of Philosophy DEPARTMENT OF MATHEMATICS CHICAGO 1906 PREftS OF The New Era printing Company lancaster, pa. ON THE RESOLUTION OF HIGHER SINGULARITIES OF ALGEBRAIC CURVES INTO ORDINARY NODES. Introduction. Every algebraic curve C can be transformed by a birational transformation of the plane (a so-called Cremona transformation) into a curve C which has no other singular points than ordinary * mul- tiple points. This fundamental theorem was established by M. NoETHERf in 1875. By a birational transformation of the curve (a so-called Riemann transformation), the curve C f can further be transformed into a curve C" which has no other singular points than ordinary * double points. It seems that this theorem was first enunciated by Halphen. J Three essentially different methods have been employed for its proof: 1) Picard § uses a quadratic one-to-three transformation of the plane. 2) Bertini II uses, for the same purpose, a cubic one-to-two trans- formation of the plane. * By an ordinary multiple point, we mean, in the seqnel, a mnltiple point all of whose cyoles are linear and have distinct tangents. For the definition of cycles, see Halphen, "Traite de G4om4trie de Salmon," p. 541, and Jordan, "Cours D'Analyse," t. I, p. 563. t Noktheb, u Ueber die singnlaren Werthsysteme einer algebraischen Function nnd die singnlaren Pnnkte einer algebraischen ourve," Mathematische Annalen, Bd. 9, p. 166. Compare also, Picard, "Traite D' Analyse," t. II, p. 364, Jordan, " Conrs D' Analyse," 1. 1, p. 588, and Halphen, " Traite* de G6ome*trie de Salmon," p. 630. X Halphen, ' ' Traite de Geom^trie de Salmon, » » pp. 630, 631, 632. His proof is, however, incomplete, as has been pointed ont by Picard, "Traite D'Analyse," t. II, p. 366. § Picard, "Traite* D'Analyse," t. II, p. 366, 1893, and Simart, "Snr un the\)reme relatif a la transformation des courbes algSbriques, " Comptes rendus de VAcademie, t. 116, p. 1047, 1893. || Bertini, " Trasformazione di una cnrva algebrica in nn'altra con soli pnnti doppi," Mathematische Annalen, Bd. 44, p. 158, 1894. 3 166202 4 SINGULARITIES OF ALGEBRAIC CURVES. 3) Polncar£ * transforms the plane curve into a curve in space, which has no singular points ; and then shows that this curve may be projected into a plane curve with no other singular points than ordinary double points. All these proofs, however, are written in an exceedingly concise style and leave a great many minor points to the reader. The object of the present paper is to give a new proof of the theorem in question by carrying out, in detail, an idea contained in the following foot-note added by Professor Felix Klein to Bertini's paper : " Die Methode von Bertini kommt geometrisch zu reden darauf zuruck, die Ebene, in welcher uns die Curve mit siDgularen Punkte gegeben ist, als eindeutige Abbildung einer Flache 3. Ordnung zu betrachten, dadurch die Curve in eine Raumcurve zu verwandeln und letztere hinterher wieder von einen hinreichend allgemeinen Punkte aus auf eine andere Ebene zu projiciren. In dieser Form ist mir der Ansatz noch von Clebsch her bekannt, der mir denselben in Herbst 1869 mundlich mittheilte." The paper is divided into four chapters. In Chapter I, those prop- erties of triply infinite systems of cubics are established, which are needed for the transformation from the plane to the cubic surface. Chapters II and III deal with the transformation from the plane to the cubic surface, f and its effect upon the given algebraic curve. Let A 19 A 2 , • • • , A r be the singular points of the given algebraic curve -ST in the original plane II ; they are all supposed to be ordinary multiple points. It is proved that the six fundamental points of the transformation can always be so chosen that one of the multiple points of K, say A l} is resolved into simple points of the image K' of K on the cubic surface and all the remaining multiple points A 2 , • • •, A r of iTare transformed into new ordinary J multiple points *Poincabk, "Sur les transformations birationelles des courbes alg^briques," Comptes rendus de VAcademie, t. 117, p. 18, 1893. For additional literature see " Encyklopadie der Mathematisehen Wissenschaften," Band III, , Heft 3, 1906. fClebscb's well-known one-to-one correspondence between plane and cubic sur- face ("Geometrie auf den Flachen dritter Ordnung," Crelle, 65). t Each oneof these points isthecenterof a finite number of linear cycles (Jordan, "Cours D' Analyse," t. I, p. 563) with distinct tangents. INTRODUCTION. 5 A' 2 j • • •, A' r of K' unchanged in orders of multiplicity and with dis- tinct tangents. In Chapter IV, the curve K' is projected into a new curve K" ', in a plane II', from a center of projection chosen on the cubic surface. It is proved that the point can always be so chosen that the curve K" possesses one multiple point less than the curve K and has all the remaining multiple points A 2 , • • •, A r of K transformed into new ordinary multiple points A' t , • • • , A' r unchanged in orders of mul- tiplicity and with distinct tangents, and a finite number of additional ordinary double points, but no other singular points. A finite number of repetitions of this process leads to a curve which has no other singular points than ordinary nodes. CHAPTER I. Properties of Triply Infinite Linear Systems of Plane Cubics. §1. Triply infinite linear systems of cubics. We shall develop, in this chapter, a number of auxiliary theorems on triply infinite linear systems of cubics, which will be needed in our discussion of the one-to-one correspondence between the plane and the cubic surface. In a plane II, whose points are referred to a system of trilinear coordinates x:y :z , there are given six distinct points P^x^ y if z.), (t a* 1, 2, •••, 6), chosen so that no three points lie on a straight line. We consider the problem : To determine all cubics passing through the six points. Let / = Ax 3 + By 3 + Cz* + Dx 2 y + Erfz + Fxtf \ + Gxz 2 + Hy 2 z + Iyz 2 + Jxyz = 0, where A, B , C, • • •, are arbitrary constants and x, y, z are homo- geneous coordinates, be the general equation of a cubic. If the cubic, /= 0, is to pass through the six points P the coef- ficients A , B , • • • , must satisfy the six equations (2) Ax 3 + By 3 + Ck\ + • • • = (* = i, 2, • -, 6). The matrix of the coefficients, m=\\x\ y 3 z\ x 2 Vi x 2 z. x.y 2 xz\ y 2 z. y.z 2 x.y t z.\\ y is always of rank six ; that is, at least, one of its determinants of order six is 4= . For, if we choose the vertices of the triangle of reference to coincide with the points P 19 P 2 , P 3 , respectively, we have and therefore, since no three of the six points are to lie on a straight 6 PROPERTIES OF LINEAR SYSTEM. line, x v straight line, , z 6 =}= ; moreover, since P 4 , P 6 , P 6 do not lie on a *4 Vi z t X 5 ft Z 5 X 6 ft \ + 0. Therefore the determinant %Va x iVl x 4Vih x ly 5 x *yl x *v*h x ly 6 %y\ *&** to which, in the present case, one of the determinants of 9ft reduces, is =f ®y an d consequently the rank of 9ft, which is invariant under a transformation of coordinates, is six. At the same time, we obtain from ( 2 ) , for this special triangle of reference, A — B — (7=0; and if we solve equations (3) Dx\y. + Ex\z. + Fx.y\ + • • • = ( t = 4, 5, 6), with respect to D, F, J and substitute the values in (1), we get, for the system of cubics, JI(a;^ + fi[xy + y[xz + yz)y + G(a' 2 xy + fi' 2 tf (4) + 7> + *)* + J«<* + P' 3 xy + Vsxz + z*)y + E{a[xy -f fi\f + 7^ + **)* = °; where a[ , £J , • • • , are fixed constants. Hence, if we denote fi s K^ + #^2/ + 7^2 + yz)y s ^y, Si = K*2/ + ^y 2 + y' 2 y z + **)* ■ / 3 = K* 2 + £3 x y + 7>2 -f 2 2 )2, = 3 y, / 4 = (a^zy + /S^i/ 2 + 7^2; + xz)x = <£ 4 x, a = H, fi= G , 7 = /, B=z E; we obtain the totality of cubics (5) 8 SINGULARITIES OF ALGEBRAIC CURVES. through the six points in the form (6) 2«*/i+>/ J +')'/, + */«-0, a, /3, 7, 8 being arbitrary constants. The four cubics/^, f t9 f i9 f 4 are linearly independent ; that is, it is impossible to determine four constants a, ft, 7, 8, not all zero, so that (7) a/ 1 + /3/ 2 + 7 / 3+ 8/>0, since each one of these four cubics f 19 f 2i f s , jf\ contains a term which does not occur in the other three ; namely, the terms y 2 z , xz 2 , yz 2 , x 2 z respectively. Hence the cubics through six points form, with- out exception, a triply infinite system, provided that no three of the six points lie on a straight line. § 2. Eeduction of the base cubics to canonical form. Instead of the four base cubics f lt / 2 , f z > f A , we introduce four other base cubics each of which breaks up into three straight lines, in the following manner : f x and f s have the common factor y, and the two conies, $ t = 0, <\> z = 0, pass through the four fundamental points P 2 , P 4 , P 5 , P 6 ; hence the pencil, afa + y 3 =0 , contains the product of the two lines, P±P 6 , P^P** > say f° r a = «', 7 = 7', and also the product of the lines, P 4 P 5 , P 2 P&> say ^ or <* = a "> 7 = 7". Hence the two cubics, break up into the lines P 1 P 3 P^- P 2 P 5 and P X P Z P 4 P 5 - P 2 P 6 re- spectively. Similarly the constants /3', 8' and /3", 8" can be so selected that the two cubics, f £-*/,+>/« break up into the lines P 2 P 3 • P 4 P 6 • P, P 6 and P 2 P 3 • P, P, • P 4 P 5 re- spectively. PROPERTIES OF LINEAR SYSTEM. Since a l mw(&-< V )mw(e i z-l3f>)mP t P t -P l P t -P t P v f t mx^ t m^(e 1 \-a 1 p)m^(fi a -^)mP t P t ■ P 2 P 3 ■ P X P„ en) \ J ^ n r K ; \f t ma* t m (12) l^"^" w(6 I X-o 1 /.)«,*( W! _ 7 ,y)«P,P,.P 1 P,.P I P,, where a 1} ■ • • , % are the minors of A corresponding to the elements « x , • • • , c 3 respectively. From the preceding normal form, the fol- lowing results are easily proved : 10 SINGULARITIES OF ALGEBRAIC CURVES. 1) The base cubics have no other common point of intersecti on except the six points P. ( i — 1 , 2 , • • • , 6 ) . 2) Not one of the points, P if is a multiple point on all the base cubics. 3) At none of the points, P if have all the base cubics coinciding tangents. (Compare the adjoining diagram.) § 3. Eank of the matrix of the first derivatives of We consider, in this section, the matrix J\x Jlx J3x Jix \ A ") Jly Jly JZy J iy J\z J-2z J$z Jiz where (14) Jix — dx > d f- f. — — J iy Qy > Jiz - dz > and propose to determine all points in the plane for which the rank, r, of the matrix is , 1 , 2 respectively. 1° r=0. When r = , all the elements in the matrix vanish ; therefore, since X f ix + yfiy + Z fi,= Zfi> the point is a common point of intersection and moreover a double point on each base cubic, against the result already established. Therefore, there are no points in the plane for which r = 0. 2° r=l. When r — 1 , all the determinants of order two vanish ; whereas, at least, one element is =f= 0. Hence it follows that at a point in which r = 1 , the four base cubics have a common tangent, since, at such a point, f . f . f _. f . f . f . ' ix * J iy * «/ iz J jx * J jy * 'jz f PROPERTIES OF LINEAR SYSTEM. 11 combining this result with the remark at the end of § 2 , we see that, at a fundamental point, the rank r can never be = 1 . The points, where r = 1 , are found by equating to zero all determinants of order two. Consider first the determinants (***) fix JZz ~J\z Jte ~ ^ f fix Jzy J\y J Sx = ^ > J \y J 3* """./ Is J Zy = " * Therefore (16) where a** *,■ ^ 2 , © 3 , of the pencil of conies (18) *4>i + 7tf>3 = °- The conies, <^ 1 = 0, <£ 3 = 0, pass through the points, P 2 , P 4 , P 5 , P 6 ; and the double points, co lf 3 , of the pencil, are the three diagonal points of the complete quadrangle P 2 P A P 5 P 6 . Consider also the determinants V A *V J2yJiz JZzJiy == ^ f J2yJix J2xJiy = " > J 2xJ 4z JlzJix "* ^ * Therefore (20) 12 SINGULARITIES OF ALGEBRAIC CURVES. From these equations, we have either x = 0, or simultaneously, (2i) M* - K+* = °> K**> -*A-°. **** - KK = °; and the points, which satisfy these equations, are the three diagonal points, ft)j , a> 2 , «3 , of the complete quadrangle Pj P 4 P 5 P 6 . It is easily proved : 1) The points, m l3 co 2 , g> 3 , do not lie on the line, x = 0. For, if we suppose one of the points, say o> 1 , the intersection of the lines, P 2 P 6 , P 4 P 5 , to lie on the line x = , which passes through the points, P 2 , P 3 , then the two lines P 2 P 3 and P 2 P 6 would contain the same point 2 , g> 3 , do not lie on the line, x = . 2) Similarly, the points, 2 , g> 3 , do not lie on the line, y = 0. 3) None of the points, a> lt a> 2> g> 3 , coincides with any of the points, •!,«£, »;. If we suppose the point, c^, to coincide with any one of the points, say (d[, then the two lines, P 2 P 6 , P 5 P & , would contain the same point, a>J, and would coincide ; hence, the three points, P 2 , P 5 , P 6 , would lie on the same straight line. In like manner, it follows that the point, (a l , does not coincide with either 3 . Simi- larly, we find that neither one of the points, o> 2 , o> 3 , coincides with any of the points, (o[ , © 2 , a> 3 . 4) The lines, x = 0, y = 0, meet at the point, P 3 ; the point, P 3 , is therefore the only point in which the six determinants, (15) and (19), vanish. But we have already shown that, at a fundamental point, not all the determinants of order two can vanish ; therefore, there are no points in the plane in which r = 1 . 3° r = 2. When r = 2 , all the determinants of order three vanish ; whereas, at least, one determinant of order two is =4= . Such points are the common points of intersection of the Jacobians of the four nets of cubics, (22) aA + /3/ 2 + 7 / 3 = , a'f, + Pf t + B'f t = , PROPERTIES OF LINEAR SYSTEM. (23) !* ly \z ♦* 3y $3* 4>4x *y 4z yK *1 + #iv yK vK 4>3 + ytly y+* * + #., x K ^4. (26) J- 2 = |y Jay + 2<#> 4 #1. , 6 and c, in ( 11 ) and ( 12 ) , f x , / 2 , / 3 , / 4 undergo the substitution {fif z ){J 2 fi)- Therefore, we obtain for the Jacobian J 4 , J Sx J 3y J Zz J\x J\y JU J2x Jly Jlz (28) Ji? (29) + a 2 Aa5/*v(c 1 X — a^) — b 2 fi 3 Axy\v'] = The Jacobians, J 2 , J 4J have the line, y — , as a common factor and intersections at the points, P 17 P 2 , P 3 , P 4 , P 5 , P 6 , a^, © 2 , a> 3 , two further points, (^ , Q 2 , and only at these points. For, if we set (30) J, = - 3yf 4 = - 3?/*% = °, (31) 14 SINGULARITIES OF ALGEBRAIC CURVES. where * Xa =(6 1 X~a lM )(c 2 X-a 2 ^)(6 2 X-a 2 /x)(c 3 /i~6 s z/) -|- ^AxfMv^X — a x n) -f c 2 y 3 Axy\fi + a 2 Axfiv(y 2 z - 7 s y) -f c t %JSxtfkfi, -f ^Ax/jlv^X — a x v) — b 2 fi^AxyXv ■ (£ 3 2/ - P 2 z )(Pi z ~ Ps x )(%v - 7 1 ^)(« 1 2/ - «»«) 4- a 2 Axfiv(P 5 y - $ 2 z) - b 2 P s AxyXv, (32) then the points of intersection of -^ 2 = and ^r 4 = can be deter- mined as follows : 1 ° /i = and v = intersect at the point P 4 . 2° /i = and % 2 = intersect at the points, P 4 , g> 3 , and P 6 , P 6 , since the point P 6 is a double point on % 2 = . 3° v = and % 4 = intersect at the points, P 4 , ©j, and P 5 , P 6 , since the point P 5 is a double point on ^ 4 = . 4° % 2 = aud % 4 = intersect at the points, P lf P 2 , P 3 , P 4 , P 5 , P 6 , o) 2 , and at the two points, Q 19 Q 2 , determined by the equations, (33) Each of the points, P 5 and P 6 , counts for, at least, two points of intersection, since P 5 is a double point on % 4 = , and P 6 is a double point on^ 2 =0. The point P 2 is a double point on both % 2 = and ^ 4 = 0, and counts therefore, at least, for four points of intersection. At P 3 , the curves, % 2 = , % k — , have coinciding tangents, whose equation is (34) x(a 2 Ab z c s + fl^x) - S^ft 1 ?! = °- At jP 1 , the equations of the tangents to %k — > X2 = are > \c PROPERTIES OP LINEAR SYSTEM. respectively, (35) Aa, &„y ( 1 — ob~c, ) — #,3 = 0, where o == ~ — — r — jr twO -^i c 2 )- , y 2 2 = o, and because (36) 7,(1 -A c i) 7 = 0, the two tangents coincide. Hence each of the points, P 3 and P 1 , counts, at least, for two points of intersection. Counting each point of intersection with the minimum multiplicity thus determined, we obtain a total of sixteen points of intersection, the exact number of 'points of intersection of the two biquadratics, %2 — ? % 4 = 0. Hence, the two biquadratics, ^ 2 = , j£ = , can have no other points of intersection than those enumerated above ; and the Jacobians, J 2 , J 4 , have the line, y = , as a common factor, and intersections at the points, P l3 P 2 , P 3 , P 4 , P b , P 6 , co iy a> 2 , a> 3 , Q l , Q 2 , and only at these points. If we interchange x and y , subscripts 1 and 2 , in (11) and ( 1 2 ) , f x , / s , f M ,f 4 undergo the substitution (f f 2 ) (/ 3 f 4 ) . There- fore we obtain, for the Jacobians, J x , J 3 , J 1 = 3xv[(b 2 \ — a 2 u)(b l X — a 1 p)(c l \ — a 1 v)(e i fi — b^v) I ' —a l Ayuv(b 2 \ — a 2 u) + c l y 3 Axy\p]=0, / 3 = _3a;/i[(c 2 X-a 2 i/)(6 1 \-a 1 /x)(c 1 X-a 1 i/)(c 3 /x-6 3 z/) ^ ' — a l Ayuv(c 2 \ — a 2 v) — b^/S^kxyXv] = 0. The Jacobians, J 3 , JJ, , have the line, x = , as a common factor, and intersections at the points, P l9 P 2 , P 3 , P 4 , P 6 , P 6 , w[,(o 2 , o> 3 , Q[ , Q 2 , and only at these points ; Q[ , Q 2 , are on x == ; and since none of the points, (*>[, co' 2 , o> 3 , Q[ , Q 2 , coincides with any one of the points, m ls a> 2 , © 3 , Q l9 Q 2 , we have the lemma : There are six, and only six, points in the plane II in which r = 2 ; namely, the six fundamental points, P i , and r = 3 , AT all other POINTS. CHAPTER II. The One-to-One Correspondence Between the Cubic Sur- face and the Plane. § 4. Transformation from plane to cubic surface. As in Chapter 1 , let P v ■ • • , P 6 be six distinct points in a plane II , no three of which lie on a straight line, and further let (1) /,(*.*»• *)**0, f,(*,l*,v)-0, where X, /a, i> denote any system of trilinear coordinates, be the four base cubics of the triply infinite system passing through the six points. Then, if x 1 :x 2 :x s :x A denote tetrahedral coordinates of a point in space, the equations, (2) m—/u ^2=/ 2 > (*****/*> P*A=f» define, in parameter representation, a cubic surface i^ 3 and, at the same time, a correspondence * between the points of the plane and the points of the cubic surface. a) To any point, P , in the plane II , different from the points, jP 1? - • •, JP r , corresponds one point, P' Q , on the surface i^ 3 ; since, not all four of the quantities, f\ , f\ , f\ , f\ , equal zero, and the ratios, x 1 : x 2 : x 3 : a? 4 =/ 1 :j 2 : / 3 :/ J , have definite values ', f\, f\, f\, fl are the values of the functions, fufzi fs> ft> respectively, at the point P . The six points, P lt • ••, P 6 , for which the functions, f lf f 2 , f if f 4) vanish simulta- neously, are called fundamental points. b) To a fundamental point, together with the totality of the paths of approach to it, corresponds a fundamental curve on P 3 . * Clebsch, " Geometrie auf den Flachen dritter Ordnung," Crelle, 65. 16 ONE-TO-ONE CORRESPONDENCE. 17 Let (3) \ = \ + \t } j£ = A* +V> v=v o+ v i t > (4) 14=0/ be a path of approach, L, to a fundamental point, P . At P , /J -/!-/{ -/i.-o, but according to § 3, not all of the quantities,/^, • • •, f° 4v} = 0; Expanding the functions,^, y^,/^,/^, in the vicinity of P , we get (5) ^•=/?x(^\)+/?m(^-^o)+/>(^-^)+--- ) = 1,2,3,4), and by (3) . • . (6) /• + powers of «, ap- proach to definite values, (7) '«*-\fk + *Jli'+*Jl> and define a definite point. For, suppose (7«) \Jl + ^fl + ^n = 0, for /- 1,2,3,4. Then, since by § 3 at a fundamental point the rank of the matrix of the coefficients is two, these equations define a unique solution for \ l ifi, l :v l ; and, since V7» + Mo/?* + "of% = 3/J = o, it follows, that \ : fi Q : j/ q is a solution for the same equations ; hence, \ L : /V ^ = \ : /VV against the assumption (4). *By this notation, we mean that, at least, one of the determinants, \fi x — Pq^ , \ v i — v t\ » /"o v i — v o*"i > i s different from zero. 18 SINGULARITIES OF ALGEBRAIC CURVES. c) Two different lines of approach to P determine two different points on the fundamental curve. Let i Mi + o, be another path of approach, L' , to P , not coinciding with L, that is (9) \ K v o \ i*i Vj K K k + 0. If the two points, P^, P^', on the fundamental curve correspond- ing to the lines Z, L' , coincide, then (io) p(\fl + vji + vji) = x;/j A + /,;/^ + v[fi (i = l,2, 3,4). Therefore, (i i) (,\ - x; )/j» + ( m - k )fu + (p"i - K >/', = o • By § 3, the rank of the matrix of the coefficients in(ll)isr=2, and, by the Euler equations, we know \:/* : v is a solution of these equations ; hence, it follows, that (12) \ n v o \ Pi "l K K r = against the assumption ( 9 ). The point, P^ , describes the fundamental curve as the parameters, \ > Pi 9 v i9 ta ^ e a ^ possible values. d) A fundamental curve on F z is a straight line. For, according to § 3, the rank of the matrix, (13) \\J jk JjfJ. Jjl (.7 = 1,2,3,4), ONE-TO-ONE CORRESPONDENCE. 19 is r=2. Hence, if we suppose, for instance, that/J K /^— /5 M ./ , J V + > two factors m' and »' can be found, so that, .« = «#£ + »:#; and these values substituted in (7) give (14) axj -/£ (m'\ + /*,) + < /5J(VX I + »,)• If we set m'X + /i, = do not lie on a conic. If to a non-fundamental point, P , corresponds the same point on F 3 as to the non-fundamental point, P , then (16) A A A A J 1 «/ 2 J 3 J i = 0. The quantities, f\ , f\ , f\ , f\ , are not all equal to zero ; suppose f\ =(= ; (1 6 ) is then equivalent * to the three equations, (17) AA ~AA = o , Afi -AA = o , AA -Af - o . These cubics have either a common factor or no common factor. 1 ° A common factor must be of degree, n = 1 , 2 , 3 . 1) n=l. Let the greatest common factor L of the three cubics be of degree *Kkoneckeb, " Bemerknngen zur Defcerminanten-Theorie," CreUe 72, p. 152. 20 SINGULARITIES OF ALGEBRAIC CURVES. n = 1, in X, /*, v) then, by (17), we have (18) JWA /4/1 " ^'Xzi where % x , %2 , X3 are quadratic factors. Since no three fundamental points, P x , • • • , P 6 , lie on a straight line, then, if two of them lie on L — , must the remaining four lie on each conic, %J = 0, x 2 — 0, %^ = ; because each cubic in ( 18 ) passes through each fundamental point. Therefore % x , % 2 > X3 belong to the same pencil ; but this would involve a linear relation between fufzyfsift) which is against our hypothesis. If only one fundamental point should lie on L — , then the re- maining five fundamental points must all lie on each conic x[ > X2 > X s ; hence %[ > Xt t Xs wou ld coincide, and therefore the common factor in (18) would be of higher degree than w= 1, against the assumption. If no fundamental point should lie on the line L = , then all six fundamental points must lie on each conic, x[ sm fy X2 = > Xi ess ^1 against the hypothesis. 2) n — 2. Let the greatest common factor i/r of the three cubics be of degree n = 2, in X, /a, v ; then, from (17), we have (19) Afi-Afi-ir-h, Af>-Afx**k> \-Af t -AA-+-h> where l 19 l 2> l s are linear factors. Since all six points, P lt • • •, P 6 , do not lie on a conic, then, if five of them should lie on ty = , the remaining one must lie on each line, l x = , l 2 = , ? 8 = ; hence a linear relation must exist between l lf l 2 , l 3 , and therefore between fiyfzyfzifv against the hypothesis. If less than five fundamental points should lie on yjr = 0, then the remaining ones must lie on each line, l x = , l 2 = , / 3 = ; hence these lines would coincide, ONE-TO-ONE CORRESPONDENCE. 21 and therefore the common factor, in ( 19 ) , would be of degree n — 3 , against the assumption. In like manner, it is easily proved, if n = 3 , that a linear relation exists between f 19 f 2 , f 3 , f 4 , in which not all the coefficients equal zero. Therefore, it follows, that the three cubics ( 17 ) do not contain any common factor. 2° Since the three cubics, in (17), do not contain any common factor, they can intersect in only a finite number of points ; we know already seven points of intersection, P 0> P l9 • • • , P 6 . Suppose one additional point P , different from the seven points, P , jPj, • • •, P 6 , to be a point of intersection of these cubics, then they meet in eight points and therefore have a ninth base point in common ; hence, they belong to the same pencil and one of them is expressible linearly in terms of the other two. Set (20) P 2 (flf 2 -fifi) + p 3 (/J/ s -/S/J + P^Af-flA) = 0, where p 2) p 3 , p 4 are not all equal to zero; therefore (2i) -/, (ft/; + pji + ft/n + ft/?/, + ft/j/ 3 + ft/?/ 4 = o, since p 2 , p 3 , p i are not all equal to zero and f\ =f= 0, the coefficients of this linear relation can not all be zero ; but this is a contradiction to the previous result that /J, f 2 , f % , f± are linearly independent. b) The image of a non-fundamental point does not lie on a fun- damental line. For, if the image of the non-fundamental point, P, coincides with the point on the fundamental line corresponding to a funda- mental point, P (\, p 0> v ), determined by the linej* of approach, (22) \=\ + \t, 11 = ^+^1, v^v^+vj], then (23) \ H "o| / 1 J 2 Jz S 1 8 2 & 3 *< 8 4 * Compare \ 4, 6. = 0, 22 SINGULARITIES OF ALGEBRAIC CURVES. where (24) Sj = \fl + Kfl + "if I U = h 2, 3, 4) ; not all of the quantities S lf S 2 , S s , 8 4 are equal to zero. Suppose /Sj 4= ; then (23) * is equivalent to the three equations, (25) jfa -fA = 0, / t S s -f.S, = 0, fX CfA = °- It follows then exactly as under a) that these three cubics can have no common factor ; they intersect therefore in a finite number of points, and we know already seven points of intersection ; namely, the six fundamental points, one of which, _P , counts twice, since, at the point P , the line (22) meets each cubic in two coinciding points; for, if we substitute \, fi, i> of (22) in (25) and expand according to powers of t , the expansions begin with the second power of t. Hence we infer as under a) that the three cubics (25) can have no further point in common. c) The image of a non-fundamental point is not a singular point on the surface F 3 . Let P be any non-fundamental point and consider the line L through P , (26) \ = \ + \t, \i = fi Q + f^t, P** v + P t t 9 (27) *0 II X ^o v o Ik ft v i The image P' 9 of P is given by (28) x l :x 2 :x 3 :x i =fl--fl-fl--A, and the tangent to the image of L at P ' Q by (29) x, =f\ + V, x 2 =/» + 8 2 t, x, =/» + Bfo x t =fl + SJ. By the Euler equations, we get (30) 3/; = \„/° A +■*/£ + "of% = 1,2,3,4) and substituting (24) and (30) in (29), we get (31) P "xj=/ U\ + 3V) f&fc + 3 V) +fK"o + 3 "i0- *Kronecker, OreZte, 72. ONE-TO-ONE CORRESPONDENCE. 23 Set X -J- 3X^ = X', a* + 3a*! t = a*' , v Q + 3^ = v , therefore (32) Ay = ^ + ^ + ^J. Since, according to §3, the rank of the matrix, \\f%f% f Q j V \\i *s r = 3 , the equations ( 32 ) represent, in parameter representation with the parameters X', a*', v , a fixed plane through P ', independent of Xj, a*i, v l9 and the tangent (29) lies in this plane. The surface F 3 has therefore atPJa determinate tangent plane and the point P' is a non -singular point. d) Two fundamental lines on the surface F 3 have no common point. If the fundamental line L[ corresponding to P x and the funda- mental line Z 3 corresponding to P 3 have a common point P' deter- mined on L[ by the path of approach to P 1 ( 1 , , ) , (where X, a*, v are trilinear coordinates with respect to the triangle of reference P l P 2 P i) these three fundamental points are its vertices) (33) x = i+x;*, ^ = o+a*;*, p=o+vit 9 10 K f*[ v [ + o, and on L' 5 by the path of approach to P 3 ( , 0, 1 ) , (34) then (35) where (35.) \ = o+v, /t = o+/« 1 <, 1 1+ Vl t, + o, Sf Sf #£> Sfl «<» -Sw Sg> fl»| = °' ' f»„( d Ji\ ... f<»={?£\ 24 SINGULARITIES OF ALGEBRAIC CURVES. By the canonical form of f 19 f 2 , f 3 , f 4 , we find, (35,) I ^ ( 3 l) - ^; c i7 3> £?> - -K c i , y 3 +^ 1 7 2 . Since not both \ , /^ equal to zero, suppose p t 4= , therefore >S' 1 (3) =j= ; then (35) is equivalent to the three equations, (36) S^Sf-SfS^O, 8p8®-8p8n**0 9 Sf S?-S*> Ajf>-0. Therefore, (^ftft - \PA)ti - Pgfiifal = 0, I ; (37) . (ta/ta-tatoiMr*! n ; .(\^^7rAMA*)K + ^ft^i"'' IIL In I and III, if ^ = , then ^ = , since /^ 4= ; against the assumption in (33) ; therefore, we have fi[ 4= 0. In II, since fi[ 4= 0, we have (38) &A/*i*t-^«bft7i" d x 6 X a 2 Cj 6 X c 1 «2 6 2 a 2 C 2 6 2 C 2 «3 6 3 a 3 C 3 6 3 C 3 = 0, the necessary and sufficient condition for all six fundamental points to lie on a conic, against the hypothesis. In like manner, it follows, if fi l = > that all six fundamental points lie on a conic. What has just been proved for the two fundamental points P t , P 3 holds for any two fundamental points. Hence two fundamental lines on the surface F 3 have no common point As a result of these theorems, it follows, that to every point of F 3 , not on a fundamental line, corresponds one, and but one, point in II, without exception ; and to every point of F z , on a fundamental line, corresponds, in II , one, and but one, fundamental point together with one, and but one, path of approach to it, without exception ; ONE-TO-ONE CORRESPONDENCE. 25 there are, therefore, no fundamental points on F 3 and no fundamental curves in the plane II. § 6. Determination of all straight lines on jP 3 . Let 1 fa + 2 * 2 + £ 3 tf 3 + £A = 0, where (40) + 0, #1 ^2 03 ^^ be any straight line, I! , in space different from one of the funda- mental lines ; the parameters A, /x, v of its points of intersection with the surface i^ , are the coordinates of the points of intersection, different from the fundamental points, of the two cubics a i/i + a J% + a Ji + a Ji = °> 01^ + 02/2 + 03/3+04/4=0, (41) in the plane II . If the line, X', is to lie on the surface F it these two cubics must have an infinity of points in common ; they must therefore have a common factor of degree, n = 3 , 2, 1 . 1) Let be a common factor of degree, n = 3, in \, ft, v . Set I 0x/ + 2 / 2 + 03/3 + PJ* = P*4>,Pi * 0; therefore (43) foa, - *£)/, + foa 2 - p^,)/, + (p 2 « 3 - ^^)/ 3 + (^2«4~/ 3 i04)/4=O, which is a linear relation, in which on account of (40) not all the coefficients vanish, against the hypothesis. 2) Suppose the greatest common divisor \jr of the two cubics to be of degree, n = 2, in X, fi, v. Set Wi + 2 / 2 + 3 / 3 + 04/ = +' A> where L x , L 2 are linear factors. 26 SINGULARITIES OF ALGEBRAIC CURVES. Now the two cubics pass through the six fundamental points, which do not all lie on a conic. Hence the conic, i/r = , can pass through at most five of the six fundamental points. On the other hand, not less than five of the fundamental points can lie on ty — ; for, if there were four, the lines L Y , L 2 would pass through the two remaining points and would therefore coincide, and the two cubics would have a common factor of degree, n — 3 ; if there were less than four, three or more fundamental points would lie on a straight line. Hence ty must pass through exactly five of the fundamental points. Conversely, if ^r = be any conic passing through five of the six fundamental points, and L x , L 2 are two distinct straight lines inter- secting at the sixth fundamental point, then the two cubics, sfr • L x = , sjr ■ L 2 = , belong to the linear system of cubics through the points, Pj , • • • , P 6 , and are therefore expressible in the form ( 44 ) . To the points of the conic sfr = corresponds then a straight line on the surface P 3 . Now six, and only six, different conies can be described to pass through five, at a time, of the six points, P Y , • • • , P 6 ; hence the conic i/r = can occupy six different positions and corre- spondingly, we obtain six straight lines on the surface P 3 . 3) Let the greatest common factor of the two cubics be of degree, n = 1 , in X, //, v. Set f «i/i + C, D satisfy besides (12) the equation (14) AK + BL r + CM r + JDN r = 0, where (15) K L l ^'0 *o, then we can solve equations (12) and (14) for two of the quantities, say A and B, in terms of the others, C and D ; hence, by substi- tuting the values thus obtained in (10), we get a pencil of planes, whose common axis is given by the equations (16) x l K K X 2 h 4 = 0, X S x. M r K, K r K L r N = 0; and, in which case, the line (16) has, at P' Q) more than r coinciding points in common with the cycle (C). The number r is called the order of the cycle, and the line (16), in parameter representation (17) is the tangent to the cycle. px 2 =L + L r u; px 4 = N + N r u, 30 SINGULARITIES OF ALGEBRAIC CURVES. § 8. Transformation of the non-singular points of the algebraic curve k to the cubic surface. We now apply the results of the preceding sections to the resolu- tion of the singular points of an algebraic curve into double points. Let (18) K: G(\ ff i,v) = 0, be the algebraic curve in the plane LT ; whose singularities we wish to resolve. We suppose that the curve K is irreducible and that it has r ordinary multiple points, A lf • • • , A r , with only linear cycles and with distinct tangents, and no other singular points.* We may further suppose without loss of generality that the degree of the curve K is greater than three. We let one of the six fundamental points coincide with the multi- ple point, A lf to be resolved, but choose all other fundamental points external to K. The image of if is a curve K' on F 3 ; we propose to study the properties of this curve K\ Let P' be any point of the curve K '; according to § 5, P' is the image of one, and but one, point P of K. The point P is either an ordinary point, or one of the multiple points, A 2 , • • •, A r> or the multiple point A x . If P is an ordinary point, then all points of K in the vicinity of P are represented by one linear cycle ( C) , />\ = \ 4- V + V 2 + •••> PI* = 1*0 + ^1 + /*,* + . .., (C) pv=Vt + Vl t + v 2 t 2 + -.., where not all of the quantities X , fi Q , v Q equal zero and (19) (20) /*n + 0. (21) The image of the cycle (C) is a cycle (C), [ P 'x l = K, + K l t+K 2 ? + p'x 2 = L Q + L 1 t + L 2 t 2 +. p'x z = M (i + M l t + M 2 t 2 + p'x i = N + N 1 t+N 2 t 2 +. (C) *Noether, Mathernatisehe Annalen, Bd. 9, p. 166. TRANSFORMATION OF AN ALGEBRAIC CURVE. 31 a cycle of the curve K' ; where A =fl A = \/» + Kfl + *J%% ■ (22) Since, according to our choice of the fundamental points, JP is in the present case not a fundamental point, not all of the quantities, K Q , A» &., N , equal zero. Suppose iT 4= 0. We propose to prove that the cycle (C) is again a linear cycle. For if (23) we should have M iV K. L o IsT, Z, J/, JV, = 0, (24) JT^p"^, A -/A. Xx>-P"X*> N^p'N,, or by applying Euler's theorem, (25) ( p\ - X x )fl + (p> - j^)/*, + (p"„ - ^/J, « (.7 = 1,2,3,4). The rank * of the matrix of the coefficients in these equations is r = 3 ; hence, (26) p\-\ l = 0, p"^-^ = 0, p\- Vl =0, therefore, (9 <) X i /*i v i = 0, against the assumption that (C) is a linear cycle. Therefore, we must have (28) K L M, iV || K, L t M x JV, + 0; *See§3. 32 SINGULARITIES OF ALGEBRAIC CURVES. that is ( C ) is a linear cycle of the curve K' ; hence the image JP' of an ordinary point P of K is an ordinary point of K' . § 9. Transformation of the multiple points A 2 , • ■ • , A r to THE CUBIC SURFACE. If P is one of the multiple points, A^ , • • •• , A r , of order of multi- plicity +..., where F| = \;/J A + K/{„+ v[fl, etc. Suppose (C) and (C[) have coinciding tangents ; that is (32) ( X iM 1 [/3 1 7 2 (>«i« 2 - A*i«i) + W0 8 y»*i - ^i7s^)] + ^l7 2 63 C 3( X 2^1 - \^ 2 ) = °J Since we supposed i£i 4 s , therefore, /a x + . In IT, if \ = 0, then p, = 0, against the assumption 1 4= ; hence, \ 4= (44) I; II; III. (45) Mi By equating the values of \/* 2 — X^ obtained from I and III, we have (46) V 2 /3 2 7 2 - M 1 (a,6 3 c 3 A + /S,?,*,) = 0; and by II , therefore \(a 2 Ab 3 c 3 + ^7,0,) - fi^ff^ = 0; a 2&2% «l^ C 3 A + ^2 7 2 a i a 2 6 3 C 3 A + a 2^l7i O^Ti ^^l^l + Oj^^^a + «1«2 6 3 C 3 A - °> (47) hence (48) 0, therefore (49) and finally, (50) a x 6 X a x Cj 6j c x «2 6 2 a 2 c 2 M 2 «3 6 3 a 3 C 3 6 3 C 3 o, 36 SINGULARITIES OF ALGEBRAIC CURVES. the necessary and sufficient condition that all six fundamental points shall lie on a conic, against the hypothesis. Hence, the cycle ( C ) of the curve K' is a linear cycle. According to § 4, the centers of the £='i£ > PV = t lVoy p^=t^ Qf /jt = ^t -}-^. The coordinates of the projection, P' ' , are (2) />£ = £o> PV = V , P?- $» t = 0. The quantities f , rj Q , f , which are not all three zero, are at the same time the trilinear coordinates * of the point, jP' ', with respect to the triangle of reference ABC in II'. Only when the point, P'q, coincides with does its projection become indeterminate. In order to avoid this exceptional case in the projection of K\ we impose, upon the point 0, condition I: that the point shall not lie on the curve K' . * Clebsch-Lindemann " Vorlesungen uber Geometric," Bd. II, p. 99. 37 38 SINGULARITIES OF ALGEBRAIC CURVES. Vice versa : If £ , n Q , f are the trilinear coordinates of any point PJJ in II' with respect to the triangle of reference ABC, then the tetrahedral coordinates of the point PJ, are ?o > v o i ?o ? ^ an ^ ^he P ro " jecting ray OP' Q is given by the equations, (3) />? = £ n**%h> &*~ZAt pT = t 2 . If we substitute these values in the equation of the cubic surface, we will get a cubic equation in t x :t 2 . If this cubic equation does not degenerate into an identity its three roots will give the parameters of the three points of intersection of the ray OP" with F z . One of these points is , the two other points we denote by P\ and P' 2 . If the cubic equation degenerates into an identity, it is satis- fied for all values of \ :t 2 , and therefore the ray lies wholly on F s . In order to avoid the exceptional case, we impose, upon the point 0, condition II: that the point sliall not lie on any one of the twenty -seven* right lines on F z . We thus obtain a one-to-two correspondence of points between F 3 and II' ; the only exception is the point O on P 3 , to which corresponds in the plane II' the line of intersection of the tangent plane f to F B at with the plane II'. The locus made up of the twenty-seven right lines on F 3 , we denote byA,. § 12. Kelated points. If XJ, fi° v v\ and \\, /*§, v\ are the parameters of the two points P[ and P 2 on F 3 , and if P[ does not coincide with and does not lie on one of the six fundamental lines, then \° : \t*\ '• v \ are rationally expressible in terms of \\ , p\ , v\ . The coordinates of P [ , in the new system, are where f\ , f 2 , /J , f\ are the same linear combinations of f lf f 2 ,f if *See§6. f There exists a definite tangent plane to F 3 at 0, since is not a singular point of F t (see § 5). TRANSFORMATION FROM SURFACE TO PLANE. 39 f 4 as f , y, f, t are of a? t , # 2 , # 3 , z 4 ; hence the equations for the projecting ray OP[ become (5) If X , /a , v are the parameters of one of the points P ' of intersection of OP[ with JFl , then the coordinates of P' are (6) Since P' lies at the same time on OP\ it must be possible to so determine t x : t 2 : £ 3 that (7) (8) = 0, «,/;(M»^»»j)^V AiH'^*) /;(x,m,j') /;(*,>>*) /i(^c') not all of the quantities, aw, m?, >&/;<*;> *•:, <),/;(*•:> *?> o-, are equal to zero, since P\ is different from and does not lie on one of the six fundamental lines. Set/i(Xj, fx° iy vl) 4= 0; then (8) is equivalent to the two equations These two cubics can not contain any common factor, because OP' l can 40 SINGULAKITIES OF ALGEBRAIC CURVES. not lie wholly on F s , since does not lie on one of the twenty-seven straight lines of F 3 , they intersect therefore in nine points ; eight of these points are the points, P x , • • •, P 6 , (XJ, /i° lf i>J), and a point ( X° , fx Q , v° ) , which is the image of in IT . The ninth base point P° 2 (\° 2 , fi° 2 , v\ ) is uniquely determined, and therefore X° : fi° 2 : v\ are rational functions of Xj : p\ : v\ ; say, * (10) rf>x2-*(x«, #,*?), K=f(\j,K,"?), rt~#(\',rf.*0> where <£ , ^ , % are homogeneous integral functions of Xf , ft J , ^J . § 13. One-to-one correspondence between the curves K and K" . Although the projection from the point establishes a one-to- two correspondence between the surface F B and the plane II', never- theless the correspondence between the two curves K' and K" de- fined by the same projection is a one-to-one correspondence provided the curve K in the plane II is irredueible, as we have supposed. In order to prove it, we consider any point P\ of K' and ask under what conditions will the related point P' 2 to P[ also lie on K'l The curve K' , in the new system of coordinates, is given by the equations tpr=/;(X, fi,v), G(X,pL,r)~Q. If P\ and P' 2 both lie on K' , then we have (12) G(\%tf,v ti> 0> +(K A *)> x(K rt> "M - Gi(K, K, *J) = o. *By an easy limiting process it can be shown that the formulae (10) still hold for the excluded points, with the understanding that the ratios : V : X are replaced by their limits as the point ( \°, ^° , v t °) approaches, along a given path, one of the ex- cluded points. TRANSFORMATION FROM SURFACE TO PLANE. 41 Hence the coordinates must satisfy simultaneously the two equations (14) &(%,$, 4)-Q, ffiWiAO.-'O, if P' 2 lies on K' . The equation, ^(Xj, /*J, vj)= 0, can not reduce to an identity holding for all values of \J , fi% v\; for, let $' be any point on F s , different from and not on K' , if we choose for P\ , the third point of intersection of OQ' with F 3 , then the point P' 2 coincides with (J)' and therefore does not lie on K' , and hence G x ( \\ , /ij , v\ ) =f= 0. We now make use of the hypothesis that K is irreducible * ; then the two equations (15) G(\,H.,v) = 0, G 1 (\ )H .,v) = 0, have either no common factor, or the factor G(\, p, v). In case first, G and G Y can intersect in only a finite number of points ; hence, there can be only a finite number of points P\ on K' for which P' 2 falls on iT' . In case second, for all points P[ on K' the related points P 2 lie also on K' . Hence, if we can find one particular position of P\ for which P' % does not lie on K' , then we have case first. Now, the tangent plane to F s at meets K' in at least one point, say P[ , and the ray OP\ meets F z in the three points, P[ , , ; therefore the point P' 2 must be the point and hence does not fall on the curve K' , since f is not on .ST . There exists therefore only a finite number m of pairs of related points, P\ , F 2 , which lie both on K' ; the projections of two such related points coincide in a point of the curve K" and give rise to a new multiple point of K" . These m new multiple points will be denoted by D'[ y • • •, D" m . To every other point of K" corresponds one, and but one, point of K', and in this sense, the projection from the point establishes indeed a one-to-one correspondence between the two curves K' and K" . *See§8. f See § 11. 42 SINGULARITIES OF ALGEBRAIC CURVES. (16) § 14. The orders of the cycles of the curve K" . Let ^ = £o + fi* + f/ + ---, PV=V + Vit + V a * + •••, ^ T = T o + T i^ + T 2^ 2 + •••> (C ) be a cycle of the curve K' expressed in the new system of coordi- nates. Since, according to §§8, 9, 10, all cycles of K' are linear, we have (17) ^0 '0 T o £i Vi Si T i #0, where not all three of the quantities, f , t; , JJ, , are equal to zero, since the center of (OJ) is different from the point (0, 0, 0, 1), which is not on K' . The cycle (C ) is projected into a new cycle ( CJ) in II', given by the equations '/>£ = ? + £i< + !/ + ---, (18) J pv = v„ + v l t + v^+---, . P ?= ?„ + £,« + ?/ + •••. We ask under what conditions will (Cq) be of higher order than the first? This will be the case if, and only if, (19) % V fo fi ^i Si ==o. But if (19) is satisfied, we can determine two quantities k l9 k 2 , so that (20) = ^ + ^, = ^ + ^,, = 5^ + ^, and therefore the point O lies on the tangent to the cycle (Cq), TRANSFORMATION FROM SURFACE TO PLANE. 43 which may be written (21) pi = f *i + Zi l 2> Pv - %h 4- V 2 > If, therefore, we construct, in every point P' of K' } the tangent to K' , or, in case P should be a multiple point of JS7, the tangents to the linear cycles with coinciding centers at P' and denote by Q' , in each case, the third point of intersection f of the tangent with F 3 , then as the point P describes the curve K' the point §' will de- scribe a curve on F z , which we denote by A 2 . If now we impose condition III upon the center of projection O, that the point shall be external to this locus A 2 , then linear cycles of K' are projected into linear cycles of K" , and since all the cycles of K' are linear, also all the cycles of K" will be linear. §15. The projection of the multiple points A' tf •••, A' r of K'. Let (16) and ,*"■%+■«« + <<■ + be two of the linear cycles with the multiple point A\ of K' for center, then £o % £o T o ?1 ^l 4, T, and according to § 9 (C^) and (C[) have distinct tangents, that is f. % ?o T o fi ^i ?! r, (CI) (23) + 0; (24) ?1 *?! ?1 T l H=0 *See§7. f It might happen that the tangent lies wholly on the surface F 3 , in which case it would coincide with one of the twenty-seven straight lines of F 3 . But since we have already imposed upon the condition not to lie on one of these twenty-seven straight lines, no special provision is necessary for this contingency. 44 SINGULARITIES OF ALGEBRAIC CURVES. The new cycle {C'() of K" is given by the equations (25) < PV=V + v' l t + v' 2 t 2 +--, The two cycles (GJ ) and (0'/) will have coinciding tangents if, and only if, (26) f. % ?o 1, V, r, ?; n'i c = 0. This equation has a simple geometrical meaning : the equation of the plane passing through the tangents to (OJ) and (C[) is, according to §7, Z V t T (27) ?o % ?o T o £i ^i ?i T i ?i ^i Si T i now if the point lies in the plane (27), we have (28) that is i fa % ?o T o f, ^i ?i T l £ >?; s = 0, ?o % ?o fl *1 Si fi *i Si This plane (27) is the tangent plane to F s at P' ; for the tangents to the cycles (C' ) and (C[) are, at the same time, tangents to the sur- face F 3 ; and since F 3 has a definite tangent plane at A\ (see § 5), TRANSFORMATION FROM SURFACE TO PLANE. 45 this tangent plane must be identical with the plane through the two distinct tangents to the cycles (C' Q ) and (C{). The plane (27) cuts the surface F 3 in a curve H. Hence, if we construct tangent planes to F 3 at each multiple point A 2 , •* • \ A' r of K' , we shall obtain r — 1 curves H 2y •••, H r , which constitute together a locus which we denote by A 3 ; and if we impose, upon the center of projection 0, condition IV, that O shall be external to this locus A 3 , then for each one of the multiple points A 2 , • • •, A' r of K' the a. linear cycles of A\ will be projected into a. linear cycles with the same center A\ and imth distinct tangents. From the conditions I— IV imposed so far upon the point 0, it follows therefore that if P" be any point of K" different from the m points DJ', . • •, D£ f then either P" is the projection of an ordinary point of K' y in which case P" is again an ordinary point of K"\ or P" is the projection of one of the multiple points A\ (i -«■ 2, • • • , r), with ?i 9 T i ) anc * &*{ f 2 9 Vz , £j > T 2 ) tne two related points of F z , both lying on K' ', whose projections P^(|, , Vi y ?i ) ana< P'i ( &> %> (*) in the plane II' coincide in the point D", so that (29) ii Vi ^ & % f, = 0, at the same time not all three of the quantities f u ^, £, nor f 2 , ?7 2 , £ 2 can be equal to zero, since on account of condition I neither P[ nor P 2 coincides with . Further the points P [ and P J are distinct, according to § 13, hence (30) £ 'h ?i T i + 0. »2 ^2 '2 2 According to § 16 , P{ and P 2 are both ordinary points of K f ; let (31) (CD TRANSFORMATION FROM SURFACE TO PLANE. 47 and (32) >? = & + £< + pv = v 2 + v' 2 t + » PT=T 2 +T' i t + (?=?, + ?;< + (C?) (c;) respectively ; their coinciding center is the point f t : rj l : fj = f 2 : rj 2 : £ 2 . The tangents to the two cycles (C") and (C' 2 ') will coincide if, and only if, £i ** £i (35) g v [ r; =0. ?2 ^2 G This equation has a simple geometrical meaning : the equation of the plane through the tangent to the cycle (C[) and the point P' 2 is (36) f V t T *i % t, r, % v[ G *; fc % ?2 T l = 0, the condition that this plane (which on account of (29) always passes 48 SINGULARITIES OF ALGEBRAIC CURVES. through 0) contains the tangent to the cycle ((7 2 ) is £« v 2 Z T 2 fl % ^ Tj fl ^1 ?! Tj fj ^2 ?2 T 2 (37) since the point g' 2 , $j , Ja , T2 is a point of the tangent to ( C 2 ) . But according to (29), a quantity />4= can be determined so that (38) f 2 = P?i> P? a ?2 = P?1 Hence if we subtract, in the determinant (37), from the last row the second multiplied by p and notice that t 2 4= pr x on account of (30), we obtain (35). The tangents to the two cycles (C") 9 (C 2 ') with the center D" will therefore coincide if, and only if, the tangents to the two cycles (C[) and (C 2) lie in one plane passing through 0. We must now impose upon the point 0, a condition which will prevent the point O from lying in a plane containing two tangents to K' '. For this purpose, let P[ be any ordinary point of K' , and let (39) &+;fo-o represent the pencil of planes through the tangent to K' at P[ ; f , 77, being homogeneous linear functions of f , 77, £, t . Let P 2 be another point of K\ the image of an ordinary point P 2 of K\ let (40) J /i = /i + /i^+..-, I */ = *> + V + •*•> be the linear cycle which represents the curve iT in the vicinity of P 2 ; then the series (40) satisfy identically the equation of the curve K, (41) G(\,^v)m.O, TRANSFORMATION FROM SURFACE TO PLANE. 49 hence (42) G(\, AS, v t ) = 0, (43) \G X (\, im , p t ) + Pk$,(\j a* , "„) + v i G A\> *•*> v o) = °- The tangeDt to K" at P' % is, according to § 8, given by the equa- tions \i*-fl + 4t, in-fl + At, where (45) \Jj=M x *>*» v o) (i = l,2,3,4), Hence if the tangent to K' at Pg is to lie in the plane (39), it is necessary and sufficient that (46) where f\, f\, («?, «°) are the same linear functions of /J, • («}, • • •, sj) as ?, »? are of f , if, K, t. Hence we must have > ft) (47) 7^-^ = 0, or, if we make use of Euler's theorem, (/Vi-*n)(AA-AA)+(\*-*\KAA-AA) +(Vi - V.XAA -A A) = o- But the equation (42) may be written (49) \G X (\, fi 0> * ) + /*o#m( x o> ^o> *o) + "o^K* A*o> *o)*° > therefore (48) (50) 50 SINGULARITIES OF ALGEBRAIC CURVES. Hence we obtain the result : If the tangent to K' at P' 2 lies in one of the planes of the pencil (39) then the parameters \ , fi of v must satisfy the two equations = 0, J IK JZk «A (51) H(\, /*, v) m fly. fly. Gy, and f\v J2 V «V (52) G(\, M,^) = 0. Vice versa : If X , /* , v satisfy these equations there exists a plane of the pencil (39), which contains the tangent to K' at f r Since (52) is irreducible the two curves (51) and (52) have either a finite number of points of intersection, or else H is divisible by G, in which case every tangent to K' is contained in a plane of the pencil (39). In the latter case, the curve K' would be a plane curve. To prove it, we use for greater convenience non-homogeneous rectangular coordinates x , y , z, the s-axis coinciding with the tangent to K f at PJ, and origin with the point P[. If the curve K' is given in parameter representation by (53) »-*(*), y = *(0> »,***(*)» and if the tangent at the point t X-x Y-y Z-z (54) j— = r 1 m —7- , v ' x y z ' is to pass through the 2-axis, we must have (55) ^'-^'=0, and if this is true for every tangent, then we obtain, by integrating (55), (56) y = cx, where c is a constant. That is, the curve K' lies in a plane passing through the s-axis. TRANSFORMATION FROM SURFACE TO PLANE. 51 If the curve K' is a plane curve, the curve K is a curve of the system of cubics through the six fundamental points. But we have supposed that K is of higher order than the third (see § 8), hence this exceptional case can not take place, and there exists therefore only a finite number of points P^on K' in which the tangent to K' lies in a plane of the pencil (39). For each such point P 2 draw the line P[P 2 , it meets the cubic surface F 3 in a third point R. As the point P[ describes the curve K\ these points R will de- scribe a finite number of curves on F B . The totality of these curves makes up a locus which we denote by A 5 . Hence if we impose upon the center of projection O, condition VI, that O shall not lie on this locus A 5 , then the new double points D " , • • • , D " m will have distinct tangents. We will now give a resume" of our results. There was given in the plane II an irreducible algebraic curve K (of order higher than the third) with r ordinary multiple points A v , A 2 , • • • , A r . We selected the six fundamental points of the transformation, so that no three lie on a straight line and not all six on a conic ; we chose one of them at the point A 1 , the other five external to the curve K. We thus obtained a one-to-one correspondence between the curve K and its image K' on the surface F % . This curve K' , we projected from a point of the surface F z upon a second plane II' . We imposed upon the point O the conditions not to lie on certain curves, viz., K' and the loci denoted by A t , A 2 , A,, A 4 , A 5 . Then we obtained a one-to-one correspondence between the curve K' and its image K" in the plane IT', and the curve K" has, corre- sponding to the multiple points A 2 , • • •, A ri r — 1 ordinary multiple points A'l , • • ., A" r of the same respective multiplicities as A % , -,A r ; whereas, the multiple point A 1 has been resolved into a x ordinary points. Besides the curve K" has a certain number of new double points D", • • •, D" m with linear cycles and distinct tangents. The points A 2 , • • •, A!' r \ D'^ • • •, D" m are the only singular points of K". We now apply to the curve K" and to the multiple point A 2 the same process which has been applied to the curve K and the multiple point A 1 , and so on. We repeat the process as many times as there 52 SINGULARITIES OF ALGEBRAIC CURVES. are multiple points A 1 , • * », A r to be resolved, taking each time the new curve as the curve to be transformed, and finally a curve will be reached possessing a finite number of ordinary double points with distinct tangents, but no other multiple points. Combining this result with Noether's Theorem mentioned in the introduction, we have the final theorem that every algebraic curve can be transformed, by a birational transformation of the curve, into a curve which has no other multiple voints but ordinary double points. OF THE UNIVERSITY OF AUTOBIOGRAPHY I was born on the 20th of August, 1863, in Oktibbeha County, Mississippi. I attended the public schools of this county for four years, the High School for four years, and the Mississippi Agri- cultural and Mechanical College for three years, graduating in 1883 with the degree of B.S., and in 1886 received the degree of M.S. I attended the University of Virginia, the summers of 1885-86-87 ; the Universities of Gottingen and Berlin the sem- esters of 1888-89. In 1895 I entered the University of Chicago as a student of mathematics and astronomy, and studied twelve quarters. I was assistant in mathematics in the Mississippi A. & M. College from 1883-1888, Professor and Head of the depart- ment from 1888-, and Director of the School of Engineering from 1902-. I received instruction in mathematics from Professors Thornton, Schoenflies, Schwarz, Hensel, Fuchs, Moore, Bolza, and Maschke, and in astronomy from Professors Laves and Moulton. My thesis was carried on with Professor Bolza. I feel under a deep obligation to all the Professors named, but especially to Professor Bolza for the continued and varied assistance given me throughout my whole time of graduate work. Buz M. Walker. 53 CONTENTS. Introduction 3 Chapter I. Properties of Triply Infinite Linear Systems of Plane Cnbics * • 6 Chapter II. The One-to-One Correspondence Between the Cubic Surface and the Plane . . . . .16 Chapter III. The Transformation of an Algebraic Curve from the Plane II Upon the Cubic Surface . . .27 Chapter IV. Transformation from the Surface F % to a Plane IF 37 Autobiography 53 54 AN tNITIAL ^ 5 TO C S tf uu BE ASSESSED FO«FA.UUB Ehe ^ THIS BOOK ON THE DATEOU E FOO BTH X .NCBEASETO SO «NTS E O gEvENTH ^ DAY AND 1 OVERDUE.