twisted cubics associated with a space curve. ii

Twisted Cubics Associated with a Space Curve. IIAuthor(s): Louis GreenSource: American Journal of Mathematics, Vol. 63, No. 2 (Apr., 1941), pp. 352-360Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2371529 .

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TWISTED CUBICS ASSOCIATED WITH A SPACE CURVE. II.*

By Louis GREEN.

1. Considerable study has been made in recent years of the local projective properties of an analytic space curve. It is the purpose of this paper to carry the investigations still further and, in particular, to study the properties of the osculating five-point cubics of a given curve r at a fixed point 0. A familiarity with some of the results of the author's previous paper on this subject 1 is assumed in Section 2.

2. An analytic space curve r, not belonging to a linear complex, may have its homogeneous differential equations written in the form

(1.1) xIV+ ax"+ (a'-0)x'+ cx 0, (O const.= 0). (1.2) VIv+ar' + (af+0)e+ct ?O,

e represents the osculating plane of r at the point with coordinates x; differen- tiation is taken with respect to a parameter u; and a, c are functions of u. At an ordinary point 0 of r corresponding to u= in0, local tetrahedra D] {x, x', a?', /"} and D2 {, e, e", "'} exist; under suitable projective trans- formations these yield the dual Halphen tetrahedra H1, H2 and the self-dual

tetrahedron of Sannia S. Relative to H1 the non-homogeneous point equations of r are

00 00

(2. 1) y X2 + E pnxn Z - X3 + E, q,xn, 7 6

where q6 is an arbitrary non-zero constant while p., qn (n ? T7) are functions of a, c, 0, q6 and are evaluated at u = uo. In particular,2

q7= 4 /4200,

q8q5 = (42 - 45 qV0 - 360 a02)/504000 0, P =

5 (_ p2 - 60 q'0 - 360a02)/1512000 0, where

= - (/60q6) 1/3, p=lc - 9d 2- 30a".

* Received August 21, 1940. 1 " T.wisted cubics associated with a space curve," American Journal of Mathe-

matics, vol. 62 (1940), pp. 285-306. This paper will be referred to as I.

2 Cfr. I, p. 286.

352



TWISTED CUBICS ASSOCIATED WITH A SPACE CURVE. II. 353

Relative to H2 the non-homogeneous plane equations of r are

00 00

(2. 2) ny 4 + EY 7n ; - 3 + EY Knd n 7 6

where K6 = q6 while 7rm, Km are obtained from pm, qn respectively by changing

the sign of 0. Hence K7 = q7, whereas 7r7 = p'7, K8 = q8 if, and only if, 4/ = 0. The problem thus arises of determining the conditions on r' in order that

=n pm and Km qn for all values of n, either at the point 0 alone or for all values of u.

THEOREM 1. A necessary and sufficient condition that 7rn p. and Kn-qm for all n and for all values of u is that r be an anharmonic curve.3

We shall prove only the sufficiency condition. From I, p. 285, we have 00

xi- E',Ai,un/n8Jl! (i , ,4) referred to the tetrahedron D, at 0, and

4

= - Eflijxj referred to the tetrahedron Hi, the coefficients Aim, /3j not j=1

involving V. Now,

Y3/Yl (y2,/Yl) + P7(y2/yl)7 + + pn (y2/yl)n +

Yi Y3-yl Y2 -2p7Yy2n-7 *-pn-lyly2n-1 -PpY2 n+

The coefficients of AUn are now equated, giving

+2 ( ) _+2( ) -_ P7 + ( ) _-*. _ *pn_ n-l ( ) p n1,

the quantities in parentheses not involving i. Since pl7q5 is expressible ration- ally in terms of a, c, 0 and their derivatives, the same is true of pl/It8-2. Hence

pniq-2 = fn (a, c, 0) with fm rational, and from the relation between pn and 7rm

we have 7rn (_ v),n-2 - fi (a, c, - G). Now the weights of /, a, 6, c, qm, p', are

0, 2, 3, 4, n -3, n - 2, respectively. Let P be anharmonic. Then the derivatives of a and c are zero, and consequently

pn n-2 =fn (a, c, 0) = (- 1) fn (a, c, -0) - mi/m2

Hence 'm = p'n at 0 and therefore at all points of r. Similarly Km = qn.

By " duality, relative to r at 0 " we mean a correspondence between a point yi = fi (pn, qm) referred to Hi, and the plane - = fz (7r, KnM) ref erred

to H2, where fl/f, is isobaric of weight 1 - i. It was proved in I, p. 289, that the dual, relative to r at 0, of a point whose coordinates are expressible in terms of q6 and q7 alone is the polar of the point with respect to a "self- dual" quadric Q at 0 whose equation referred to Hi is

3 That is, that the coefficients a and c in (1. 1) be constants. For properties of anharmonic curves see E. J. Wilczynski, Projective Differential Geometry of Curves and Ruled Surfaces, 1906, p. 279.



354 LOUIS GREEN.

54q63 (Y1Y4 + 3Y2y3) - 378q62q7 (Y32 + y2y4) + 882q6q72y3y4

+ (54q64 - 343q73) y42 = 0.

From the method by which this quadric was obtained we have immediately

THEOREM 2. A necessary and sufficient condition that 7r. = p' and

Kn= qn for all n, at the point 0, is that the dual, relative to r at 0, of any point whatever be the polar of the point with respect to the self-dual quadric Q at 0.

The local tetrahedron of Sannia, S, with vertices 0, T, N, B has recently been studied by Fon.4 This tetrahedron was completely characterized in I

with the aid of the quadric Q. Conversely, given S, we can characterize Q as follows:

THEOREM 3. The self-dual quadric Q is the only quadric with the following two properties: (a) The edges OT, ON, BT, BN of S are rulings of Q; (b) all the rulings of Q which meet ON belong to the osculating linear complex of r at 0.

The proof is straightforward and will be omitted. The tetrahedra D1, D2 are intimately related to S. The point x' coincides

with T, a' lies on ON, and x"' lies in the plane OTB. Dually, the plane e' coincides with OTB, e" contains ON, and e' passes through T.

3. The oo 1 twisted cubics, Ta, represented parametrically by the equations

y1 - 1-at3, y2 = t, y3 = t2 y4 = t3

were studied in detail in I. Each of these cubics lies on the same quadric cone through 0, has the same osculating conic at 0, and determines the same

linear complex. With the single exception of the six-point cubic (ac 0),

each cubic of the family has five-point contact with r at 0; with the exception of the six-plane cubic (ac 2q6), each cubic of the family has five-plane con-

tact with r -at 0. The principal plane at 0 of r and T. (ac =7 0) is the

osculating plane z = 0. If a is a numerical multiple of q6, the dual, relative

to r at 0, of T, is the cubic T,, for which a + oc = 2q6. These cubics constitute merely a subset of the totality of five-point cubics

of r at 0: a two-parameter family, T.B, with parametric equations

y( - 1 + 2/3t2-_ t3, y3 t2, y2=t + /t3, y4 t3.

4 Te-Chih Fon, " Note on the projective differential geometry of space curves," Annali di Matematica, vol. 18 (1939), pp. 97-106.




All cubics T.,6 with the same value of /8 possess the same osculating conic K,6 at 0, namely (4) 4y - 3X2 - 8/y2 O Z,

lie on the same quadric cone K +6:

(5) y2 _XZ + Z2 _

and determine the same linear complex. If P8 #& 0, the cubic T,6 has four-plane contact with r at 0. Its dual,

relative to r at 0, is a cubic Uy having four-point and five-plane contact with r at 0, and possessing the parametric equations

y6)l 1 + 9a2_ YT3 y3 r2, (6) Y2T2a3 Y4=

Y2 'r + 28T3p y4 ~

r 3

Any point P on T,,, is determined by a specific value of t. Let

(X-fl (pn, q-m), /3 = f2 (pn, qm), t f3 (pn, qm),

the weights of a, /, t being 3, 2, 1 respectively. We define

a= fl (7rn, Km) T3 f2 (n, Km), t = f3 (7rn, Km)K

Then the dual of the point P is the osculating plane of Uy,

y=2q6 --/q7/3q6, 8 =--/3,

at the point where

3q6t 5 7q7t- 3q6

4. We shall assume throughout the remainder of this paper that ac/8 0. The projection of T<p upon the plane z - 0 from the point (h, k, 1) has

the equations

y X2 kx3/t + 2hx4/1- (3hlc + I + /ild + a12)x5/12

(7) + (7h2 + 2k + 2/thl _-p212 + 2akl) i6/l2

- (12h27C + 9hl + 8phlcl + /12 + 9ahl2 + a/313)X7/13 + z = 0

for the projection of r from the same point we find

8 y X2 kx3/l + 2hx4/l - (3hk + l)X5/12 + (7h2 + 2k- q6kl6l)x6/12

(8) - (12h2k + 9hl - 2q6h12 + q7kl2 - p713)x7/13 + * * z = 0.

Hence the principal plane r<, of T<p (and r at 0) has the form

(9) fly+az= 0;

I f Putting =0 we get the correct formula for the dual of a point on T,. The result given in I, p. 291, holds only when a and t are functions of q6, q7.



356 LOUIS GREEN.

the principal line Io is given by

(10) x= (3? +2 2? + aq6)/A, y 2a/, z=2/2/;

while for the principal point Pafi

A = - (3ax3 + 2#3q6 + 2a3 + 5a2q6 + 2aq6J + 2cq3q7 + 2/2P7)/#.

The principal plane 7rap meets the cone K9p8 in the x-axis and in another line rap: (11)

X a + #3)/ftn y -af/A/y

Z p

This line intersects Tap at 0 and at another point Rla: for which

L-- (a3 + 3af3)/Il.

The principal lines laf: of those cubics Tafi which lie on the same cone

K'p generate a quadric cone:

(12) 2/3(y2- xz) + 2Z2 _q6yz 0;

as /8 varies this cone envelopes a quartic cone having the equation

(13) (y2 - XZ) 2 + q6yz3 = 0.6

The principal points PajS of all Tap with fixed /8 trace a twisted cubic on the cone (12), the locus of these cubics as ,B varies being a surface of the seventh order. If a, ,B both vary, the point Rafi generates the cubic ruled surface of Calapso: (14) z2- 3xyz + 2y3= 0,7

which meets the cones K', in a family of twisted cubics. The principal line 1< lies on the cone K',3 if, and only if,

(15) a /33/q6.

This condition defines a one-parameter family of cubics Tap which we shall call self-associate cubics. The principal lines of the self-associate cubics generate the quartic cone (13), the principal points trace a curve of the seventh order, and the points R& trace a sextic curve whose parametric equations are

(16) y 3 1-3Vq6t3, y2 Vq6t5- t2t y3 t4 Y4 t6.

This sextic is the locus of the vertices of all quadric cones having seven-point

6 See I, p. 304, where this cone is defined differently. 7 R. Calapso, " Sulle superficie gobbe di terzo grade (del tipo di Cayley) legate al

punto di una data superficie," Rendiconti dei Lincei, vol. 13 (1931), p. 495. See I, p. 291 for another property of this surface.




contact with r at 0.8 The principal point of a self-associate cubic Ta"p lies on

this sextic if, and only if,

(17) 84+ + +2q7? 2flq62 + q6pl7 0.

The four points thus obtained are the vertices of those quadric cones having eight-point contact with r at 0.

5. We shall say that the cubic-T p is associate to Tae81 if

(18) a2 = al 2q6/l13, 32 = alq6//12.

Since this property is symmetric, we can speak of associate cubics. Two associate cubics coincide if, and only if, (15) holds, thus justifying the above terminology of self-associate cubics.

The following two theorems characterize associate cubics geometrically.

THEOREM 4. Two cubiws are associate if, and only if, they possess the same principal line.

THEOREM 5. Each cubic Ta,,p determines a unique quadric which contains the cubic and has seven-point contact with r at 0. Taflp is self-associate if, and only if, the quadric reduces to a cone; the vertex of the cone is the point Rapl. If T,al,1 is not self-associate, there is just one other five-p,oint cubic-the associate cubic Toa2i-o-which lies on this quadric.

The equations of the seven-point quadrics containing the associate cubics

Ta1fil and T,2 are

(19) /3 2(y _ X2) - (3,I3 + a?2 + aiq6) (y2 _ XZ) + ai/h(z -xy-q6Z2) 0;

It may be remarked that the most general seven-point quadric of r at 0, whose equation is

(20) h(y-x2) + k(y2 -xz) + (z - xy -q6z2) = 0,

is one of the quadrics (19) provided the restriction hAil # 0 is imposed. The cubics Tl,,,p (i - 1, 2) are obtained from (20) by means of the conditions

(21) a4 = /341/h, /B 2h2 + /3,(hk + 12) + q6hl = 0 (i = 1,2).

THEOREM 6. Let Ta&l and Taa2f62 be associate cubics (distinct or not). The rulings through 0 of the seven-point quadric on which the cubics lie are

8 See E. P. Lane, Projective Differential Geometry of Curves and Surfaces, 1932, p. 22.



358 LOUIS GREEN.

the lines rai (11). The harmonic conjugate of the x-axis with respect to these rulings is the common principal line of the cubics.9

THEOREM 7. Let Taipi and T,arp2 be distinct associate cubics; let X be the line joining the points ],p' and ]?a22, and let M be the pole of the plane

z = 0 with respect to the seven-point quadric Q containing the cubics. Then A is a ruling of the sui,rface of Calapso (14) and meets the x-axis at the pole of the common principal plane of the cubics with respect to the osculating null system of r at 0. Furthermore, the point M is the intersection of the line A with the common principal line of the cubics. Finally, the intersection of Q with the plane z = 0 is the osculating conic at 0 of the projection of r upon this plane from the point M.

The following observation may also be added. There are two rulings on Q which are tangent to the cubic T,1. Let these rulings meet the ruling r,a at Ai (i = 1, 2). Then the harmonic conjugate of 0 with respect to the points Ai is the point R,a,O2, on the associate cubic T<O2.

6. The cubic TO will be called singular if

(22) aZ2q6 +, (q62 + fq7) + 32(q6 + p7) =0.

It is easily seen that. T.af is singular if, and only if, its associate is, singular. Since, furthermore, equations (17) and (22) are equivalent when (15) holds, it follows that there exist four singular, self-associate cubics.

Singular cubics are characterized geometrically by the following two theorems.

THEOREM 8. A cubic Tafi which is not self-associate is singular if, and only if, it possesses the same principal point as its associate.

THEOREM 9. Let M be the pole of the plane z = 0 with respect to the seven-point quadric Q containing the cubic Tl,. Then T, is singular if, and only if, Q has eight-point contact with r at 0, and this is true if, and only if, M coincides with the pincipal point of To.

The principal lines L: of the one-parameter family of singular cubics generate a quadric cone whose equation is

(23) q6y2- 2q6xz + q7yz - p7Z2 = 0,

the lines rafi generate a cubic cone:

(24) (q6X -q7y + p7Z) (y2_ XZ) + q62yz2 = 0,

9 See E. Bompiani, " Sul contatto di due curve sghembe," 1. Accademia delle Scienze dell'Istituto di Bologna, vol. 3 (1926), p. 35, for a more general statement of part of this theorem.




and the principal points Plaa trace a twisted cubic,

(25) y, = - q6 + 3q7t - 3p7t2, y3 2q6t2,

Y2 = q6t + q7t2 - p'7t3 Y4 = 2q6t3,

which is the residual intersection, besides the x-axis, of the cone (23) and the surface of Calapso (14).

It follows from (22) that each cone K', possesses two singular cubics which will be called adjoints of one another. There are three self-adjoint cubics, determined by the equation

(26) 432q6(3q6 + P7) - (q62 + 3q7 ) 2 0.

Let T,a' and T,z o be distinct adjoint cubics. Their principal lines

lal' l,a"g determine a plane whose envelope as p varies is a quadric cone:

(27) (q6x - q7y + p7Z )2 + q3yz = 0.

This cone is tangent to the plane z = 0 along the line

(28) q6X q7y= 0 .

The lines ra' , ra, a determined by the adjoint cubics lie in a plane which for all values of ,B passes through the line

(29) q6x+P7Z 0 y.

The eight-point quadrics of r at 0 are obtained from (20) by putting kq6 = hp7 + 1q7. Hence the intersections of these quadrics with the plane z = 0 form a pencil of conics having three-point contact at 0 and passing through a common fourth point on the line (28).

7. We shall now consider briefly other special members of the two- parameter family Ta.

There are two cubics of the family which lie on the unique nine-point quadric of r at 0. Their values of a, /3 are obtained from (21) and

(30) h: k: I= (q6p7-q6q8 + q72): (P72 + q7p8 -q8p'7): (q6p8 -q7p7).

It follows from our definitions that if r and any Tap are projected upon the plane z = 0 from their principal point Pap, the curves obtained have at least eight-point contact at 0. By suitable choice of as and / we can obtain nine- or even ten-point contact for these curves. For example, the cubics Ta0 which yield, upon projection from Pag, nine-point contact with the projection of ir form a one-parameter family satisfying the condition



360 LOUIS GREEN.

(31) 16 + 2#34q7 + 2p3(q62 + P'8) + 2p2(q6p'7 + xq8) +4(x3q7(X + q6)

+ q6( + 2q6) (2 + q6) =- O.

The points of any two cubics Ta,', Ta"p on the same cone K' can be put into a 1 < 1 correspondence by making corresponding points lie on the same ruling of the cone. The harmonic conjugate of 0 with respect to corresponding points of these two cubics generates another cubic TPap for which a (c' + ?ce")/2. If, now, Ta'fl and Ta",B are adjoint cubics, then

(32) a - (q62 + pq7)/2q6.

This relation between a and 8 determines a new one-parameter family of cubics Tl2afi. Their principal lines generate a quadric cone with the equation

(33) (2q6y - q7z) (2q6x - q7y)- q63Z2 _ 0,

while their principal points trace a twisted cubic on this cone. There are three self-associate cubics in this family, but no pair of distinct associate cubics can belong to the family.

The cubics associate to the family (32) form another family determined by the condition (34) 2a2q6 + af3q7 + /3q6 ? 0.

Each cone K' contains two cubics of this family, which coincide only when ,8 = q72/8q62. Forming the harmonic conjugate of 0 with respect to corresponding points on each pair of these cubics, we arrive at another family of cubics Tafi for which (35) a, /3q7/4q6.

All cubics of this family determine the same principal plane

(36) 4q6y -q7Z 0;

furthermore, the associate of each cubic belonging to the family also belongs to the family.

GEORGIA SCHOOL OF TECHNOLOGY, ATLANTA, GEORGIA.



twisted cubics associated with a space curve. ii

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