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THE TUHOKU Mathematical JOURNAL THETUHOKU MATHEMATICAl JOURNAL Projective Description of alone Quartic Curves, . by Joanna I. MAYER, San. Jose, Calif., U.S.A. Ruth Gentry introduces her dissertation On the Fonrms of lane Quartic Curves as follows : " Many papers dealing with curves of the fourth order, or Q uartic Curves, are to be found in the various mathematical pert. dicals ; but these leave the actual appearance of the. curve. as as w hole so largely to the reader's imagination that it is here proposed. to give a complete enumeration off the fundamental forms of Plane Quartic Curves as they appear when projected so as to cut the line, at infinity the least possible number of times.... , together with evidence that the forms presented can exist". It is the purpose of this paper to show whether it is possible to derive, and how to construct, all of the forms proposed in Gentry's dissertation by the method described by Doctor H. P..Pettit in his " Projective Description of Some Higher Plane Curves(1)" . The accepted analysis of hgher singularities possiblein quartic curve is here given. A Tacnode is composed of two consecutive nodes, the tangents at a tacnode count as two consecutive double tangents. A NoJe-Cusp is composed.of one node and one cusp in consecu tive position and involves one real inflexion. The tangents t a Node-cusp• count as two (non-isolated) consecutive double tangents. An 0senode is composed of three consecutive nodes, either on a right line or on the • same parabola of curvature, A Tacnode-Cusp is composed of a node-cusp and a node, that is, of two nodes and one cusp in consecutive position. A Point of the _first kind is composed of three coincident nodes, formed by the crossing of three ordinary branches with real, distinct tangents. A Triple Point of the second kind is composed of one cusp and two nodes in coincidence, formed by the passage of an ordinary (1) The Tohoku Mathematical Journal, VoL 28, No8. 1, 2, June 1927. 2 J.L MAYER: branch through a cusp ; the tangents are all real, two of them con- secutive, forming the cuspidal tangent. A Triple Point of the third kind is composed of two cusps and a node in coincidence ; the tangents are all, real, three of them consecutive. A triple Point of the fourth kind is composed of three coin- cident nodes, formed by the passage of an ordinary branch through a conjugate point (a double point with two imaginary tangents) ; one tangent is real, and two are imaginary. Classification of Quartics. I. Unicursal Rational Quartics. A. Quartics with Three real, distinct double points. i. 3 nodes ii. 2 nodes 1 conjugate point iii. 2 nodes 1 cusp iv. 3 cusps v. 2 cusps 1 conjugate point v i. 2 cusps 1 node vii. 3 conjugate points viii. 2 conjugate points 1 cusp ix. 2 conjugate points 1 node x. 1 node 1 cusp 1 conjugate point. B. Quartics with one real and Two real consecutive double points. xi. 1 Node-cusp 1 conjugate point xii: 1 Node-cusp 1 cusp xiii. 1 Node-cusp I node xiv. 1 Tacnode (opposition) 1 conjugate point xv. 1 Tacnode (opposition) 1 cusp xvi. 1 Tacnode (opposition) 1 node xvii. 1 Tacnode (embrassement) 1 conjugate point xviii. 1 Tacnode (exnbrassemermt) I cusp xix. 1 Taconrde (etnbrn.vsernent) 1 node, xx. 1 Tacriode (isolated) 1 conjugate point xxi. 1. Tacnode (isolated) 1 cusp xxii. 1 Tacnode (isolated) 1 node. C. Quartics with Three consecutive double points. xxiii. 1 Oseriode (real) PROJECTIVE DESCRIPTIONOF PLANEQUARTIC CURVES. 3 xxiv. 1 Oscnode (isolated) xxv. 1 Tacnode-cusp. D. Quartics with Three coincident double points, i.e., with a triple point. xxvi. 1. Triple point 1st kind xxvii. 1 Triple point 2nd kind xxviii. 1 Triple point 3rd kind xxix. 1 Triple point 4th kind. E. Quartics with one real, and Two imaginary double points. xxx. 1 node 2 imaginary double points xxxi. 1 cusp 2 imaginary double points xxxii. 1 conjugate point 2 imaginary double points. II. Elliptic Quartics. A. Quartics with Two real, distinct double points. i'. 2 nodes (unipartie) or bipartie( 1) ii'. 1 node 1 cusp „"" " "„ iii'. 2 cusps iv'. 2 conjugate points unipartie or bipartie v'. 1 node 1 conjugate point unipartie or bipartie vi'. 1 cusp I " B. Quartics with Two consecutive double points. vii'. 1 Node-cusp unipartie or bipartie viii'. 1 Tacnode (opposition) „ " "„ ix'. 1 Tacnode (embrasseinent) unipartie or bipartie x'. 1 Tacuode (isolated) " " "„ III. Hyperelliptic Quartics. A. Quartics with One double point i". 1 node unipartie, bipartie or tripartie ii". I cusp „ " " " "„ „ iii". 1 conjugate point unipartie, bipartie or tripartie. IV. Non-Singular Quartics. i"'. 1 closed circuit or oval ii'". 2 ovals iii'fl. 3 ovals iv"'. 4 ovals v" Annular quartic, i.e. an oval within an oval. (1) A plane elliptic quartic can consist of one closed circuit or two closed circuits, with both double points on the same or separate circuits. Hence there is it possibility of 12 types of elliptic quartics with two distinct real double points. 4 J.I. MAYER: The method described by Doctor H. P. Pettit in his " Projective Description of Some Higher Plane Curves, is as follows : " Consider two curves in the plane , C, of order n, and U2 of order m. On a point A1 chosen at random in the plane let a line 1, be drawn intersecting the curve C1, in n points P1,P2, ...... Pn. Let these n points determine n lines g1,g2 ... gn through a second point A2. Each of the lines cuts the curve C2 in m points R1 R2, .... , Rm which determine in lines k1, k2...... km through a third fixed point A3. Thus we have through A3 mn lines corresponding to the one line 1, through A, and which cut 1, in mm points. The locus of these points 'for all possible choices of the line through A, is the curve generated. Thus we have a higher type of projectivity. ` The product of two pencils of rays projectively related by means of two base curves of order m and n respectively (as described above) is a curve of order 2mn having mn-fold points at the vertices of the two pencils(1).' " When C1 is of order 2, C2 of order 2 and A0, 0, 0), A2(0,1,0) and A3(0, 0, 1) are the vertices of the triangle of reference in the plane (homogeneous coordinates) the curve generated is an octic. It is here proposed to show how this octic can be made to degenerate into different types of plane quartic curves. Let the three fixed points A1, A2,A3 be the vertices of the triangle of reference in the plane (homogeneous coordinates). Then the pencil on A1,will have the equation (1) The curve C, of order 2 will have the equation (2) -cohere i= 1, 2, 3 and ,j =1, 2, 3. The equation of the connecting pencil on A will be (3) The curve C2 of order 2 will have the equation (4) the pullcii On A3 Will have the equation (5) Eliminating x1,x2•> from equations (1), (2) and (3) there results the relation (1 ) Tohoku Mathematical Journal Vol. 28, No. 1, 2 June 1927. PROJECTIVE DESCRIPTION OF PLANE QUARTIC CURVES. 5 (6) which expresses in in terms of k. Similarly by the elimination of X2,x3 from equations (3), (4) and (5) there results the relation (7) or expresses n as a function of m, or m• as a function of n. Equation (8) is of order 4 in m and vi. Hence by the elimination of in from (6) and (8) there results an equation of degree 8 between k and n. Then if the values of k and n from equations (1) and (5) be intro- duced in this last equation there results the explicit equation of the generated curve(1) in the following determinant form Part I. Unioursal Plane Quarties. The three double points may be all real, or two be conjugate imaginaries. The case when all three are real will be first con- sidered : in this case the double points may be all distinct ; twe consecutive ; all three consecutive ; or all .three coincident. A. Quartics with Three real, distinct double points. The general appearance of the curve depends so largely upon the nature and relative position of the tangents at the double points and those from the, double points that it is here proposed to consider the curve in . subdivisions according s the tangents at the double points are real and distinct, consecutive, or imaginary, that is, ac- em-di-noras the double points are nodes cnsps or Conjugate points. Theorem I. The curve generated is a plane quartic curve having .three real distinct dps(2) at the vertices of the triangle of reference, if A1,and A2 are on c1; and A1A3,A2A3 are tangent (1) Tohoku Mathematical Journal Vol. 23, No. 1, 2 June 1927, pp. 73-74. (2) dps=double points, dp=double point. (8) 6 J. I. MAYER : respectively to CI at A1 and A2. If A1 and •A2 are on C1,and A1A3 and A2A3 are tangent to C, at A1 and A2respectively, then a11=a22=a13=0(1 ). Substituting these values in determinant I, we have when expanded : a233x 32(b11„2x21+2b12x1x2+b22x22) +4a122b33x12x22- 4a12anx1x2x3•((b13x1+b23x2) =0. The highest powers occuring in are of the second degree; therefore A1,A2, A3 are dps(2 ). The tangents at the dps(3) are : being the terms multiplying x12,x22, x32 respectively equated to zero.
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