DIMENSIONAL SPACE.
BY W. L. EDGE Of EDINBU~Gtt.
Contents.
Page
Introduction . . . 253
The Jacobian curve with two pairs of concurrent trisecants . . . 258
The Jacobian curve with four concurrent trisecants . . . 265
The Jacobian curve whose trisecants are the edges of a hexahedron . . . 272
The Jacobian curve with a scroll of trisecants . . . 275
The net of polar quadrics of points of a plane in regard to a Segre primal 315 The freedoms of the different kinds of Jacobian curves . . . 324
Table showing the various nets of quadrics in [41 . . . 331
Introduction.
W h e n t h e p r o p e r t i e s of a g e n e r a l n e t of q u a d r i c s in [41 are k n o w n it m a y be of i n t e r e s t to discuss t h e p r o p e r t i e s of some p a r t i c u l a r i s e d nets of quadrics, a n d this p a p e r is d e v o t e d to such a discussion. T h e discussion m a k e s no claim to be exhaustive; the g e n e r a l n e t of quadrics is merely p a r t i c u l a r i s e d in different ways and, u s i n g the k n o w n properties of t h e g e n e r a l n e t as a basis, some pro- perties of the p a r t i c u l a r nets are obtained.
Some properties of a g e n e r a l n e t of q u a d r i c s in [41 h a v e r e c e n t l y been expounded~; t h e y hinge, f o r t h e g r e a t e r part, u p o n t h e J a c o b i a n curve ~ of t h e
i ~)The g e o m e t r y of a n e t of q u a d r i c s in f o u r - d i m e n s i o n a l s p a c e , . Acta mathematica 64 (1935), 1 8 5 - - 2 4 2 . T h i s p a p e r will be referred to as G.Z~.Q.
net. T h e m o r e f u n d a m e n t a l of these p r o p e r t i e s m a y be briefly r e c a p i t u l a t e d here, as we shall o f t e n have to a p p e a l to t h e m . T h e p o l a r solids, in r e g a r d to t h e m ~ q u a d r i c s of t h e net, of a n y p o i n t P of 3 have in c o m m o n n o t m e r e l y a line b u t a plane, a n d this plane is a secant plane of 3, m e e t i n g it in six points; we say t h a t t h e secant plane is conjugate to P. T h e r e is t h u s a singly- infinite f a m i l y of s e c a n t planes of 3 : t h e y g e n e r a t e a locus R~ ~ on w h i c h 3 is a sextuple curve. E v e r y plane w h i c h has five i n t e r s e c t i o n s w i t h 3 m u s t have a sixth i n t e r s e c t i o n w i t h 3. A n y solid passing t h r o u g h a s e c a n t p l a n e of 3 meets 3 f u r t h e r in f o u r points a n d these f o u r points, t o g e t h e r w i t h the p o i n t P of 3 to which t h e s e c a n t plane is c o n j u g a t e , f o r m a set of five points on 3 w h i c h are the vertices of five cones of t h e n e t all b e l o n g i n g to t h e same pencil of quadrics (G.N.Q. w167 5--6). N e x t : 3 has t w e n t y t r i s e c a n t s ; t h r o u g h e a c h t r i s e c a n t t h e r e pass t h r e e secant planes of 3, a n d t h e t h r e e points of 3 to w h i c h t h e s e t h r e e secant planes are c o n j u g a t e lie on a second t r i s e c a n t ; the t h r e e s e c a n t planes which pass t h r o u o h this second t r i s e c a n t are c o n j u g a t e to t h o s e t h r e e points of 3 which lie on the first t r i s e c a n t . T h e t w e n t y t r i s e c a n t s t h u s consist of ten conjugate pairs: when two t r i s e c a n t s of 3 are c o n j u g a t e any p o i n t on e i t h e r of t h e m , w h e t h e r on 3 or not, is c o n j u g a t e to e v e r y p o i n t on t h e o t h e r in r e g a r d to e v e r y q u a d r i c of t h e net (G.N.Q. w IO). A secant p l a n e a which passes t h r o u g h a t r i s e c a n t t of 3 m e e t s 3 f u r t h e r in t h r e e points w h i c h are n o t on t, a n d the cones of the n e t whose vertices are at these t h r e e p o i n t s belong to t h e same pencil. T h e vertices of the r e m a i n i n g two cones of this pencil are on t h e t r i s e c a n t t' which is c o n j u g a t e to t; t h e y are t h o s e two of t h e t h r e e i n t e r s e c t i o n s of t' w i t h 3 o t h e r t h a n t h a t i n t e r s e c t i o n to w h i c h the s e c a n t plane a is c o n j u g a t e .
I t was s h o w n (G.N.Q. w 34) t h a t the e q u a t i o n s of t h r e e l i n e a r l y i n d e p e n d e n t quadrics in [4] can, in general, be r e d u c e d s i m u l t a n e o u s l y to t h e c a n o n i c a l f o r m
where
al X f + a~ X ] + a~X~ + e Z 2 + bl Y~ + b.~ Y~ + b~
ai X~ + a'~X 2 2 + a3 ' X ~ 3 + c . Z ~ + bl Y ~ + b~ Y ~ + b a . . .
pp p ! ip 2 pp i t pp
al X~ .q- a2 X g -b a3 X3 q- c" Z ~ q- bl Y~ + b2 y 2 ._[_ b3
x l + + x . - z = - YI + r . + Y3.
~ 0 ~ Y-~ ~ O~
H e r e a f o r m of specialisation at once leaps to t h e eye. T h e seven l i n e a r f o r m s w h i c h occur are such t h a t six of t h e m m a y be supposed to r e p r e s e n t a r b i t r a r y
solids while the seventh, Z, represents the solid which joins the line of inter- section of X l = o , X ~ = o and X 3 = o to the line of intersection of : Y I = o , Y ~ - - o and Ys = o . I f then we specialise the canonical form I by omitting the terms in Z s and consider the net of quadrics
where
al X~ + as X~ + as X] + a~ X~ + a5 X] + as X~ = o, bl Xf + b~ X~ + b3 X] + b4 X] + b~ X~ + b6 X~ ---- o, C~X, ~+ c2X~+ c~X ~
+c~X~+ c~X~+c6X~=o,
x , + ~ + ~ + ~ + ~ + ~ o
I I
and we have written - - X 4 , - - X s , - - X 6 for Yl, Y~, Y3 respectively, we have a net of quadrics related symmetrically to the six faces of a hexahedron. This however, although an admirable illustration, is somewhat too drastic a specialisa- tion to begin with; there are other specialisations of I which are not so parti- cularised as II. On the other hand there are nets of quadrics whose properties we shall consider t h a t are even more particularised than II.
In order to keep the work within reasonable compass we must impose some limit on the extent to which we specialise, and we therefore decide t h a t no consideration will be given here to any net of quadrics whose Jacobian curve breaks up into separate component curves or whose Jacobian curve has a multiple point. All the nets of quadrics t h a t are considered have Jacobian curves t h a t are in birational correspondence with plane quintic curves, of genus 6, without multiple points.
W h e n the properties of a general net of quadrics in [4] were obtained the birational correspondence between the Jacobian curve and a plane quintic was discussed in some detail, and it is of interest to see, when the net of quadrics and its Jaeobian curve are specialised in any way, how the plane quintic is specialised correspondingly. For example (cf. G.N.Q. w 27 and w 35): the equa- tion of a plane quintic in birational correspondence with the Jacobian curve of I is
~--1 + ~--1 + ~-1 + VX--1 + ~--1 + ~]~-1 _~. L (~-1 + ~-1 + ~:1 )(V-~-I + V.~-I -4- ~-1 ) = O, where
L~cx+c'y+c"z.
Hence, putting c = c ' - ~ c " = o , it follows t h a t a plane quintic in birational correspondence with the Jaeobian curve of I I has an equation~T 1 + ~T 1 + ~T 1 + ~T 1 + ~T 1 -~ ~ v l = o,
a n d is t h e r e f o r e c i r c u m s c r i b e d to t h e h e x a g r a m f o r m e d by t h e six lines ~ : o.
T h u s t h e p a r t i c u l a r i s e d q u i n t i c curve has an inscribed h e x a g r a m , w h e r e a s a g e n e r a l quintic curve has not. I t will also be s h o w n t h a t the converse of this r e s u l t is t r u e in t h e sense t h a t if a plane q u i n t i c has an inscribed h e x a g r a m it can always be p u t into b i r a t i o n a l c o r r e s p o n d e n c e with t h e J a c o b i a n curve of a n e t of q u a d r i c s given by t h e c a n o n i c a l f o r m I I .
I f t h r e e conics are t a k e n in a p l a n e a n y one meets a n y o t h e r in f o u r points, so t h a t t h e r e are twelve i n t e r s e c t i o n s in all; t h e r e are oo s q u i n t i c curves passing t h r o u g h t h e s e twelve points. Such a q u i n t i c c u r v e has two f u r t h e r in- t e r s e c t i o n s with each conic, a n d if we suppose t h a t , on each conic, these t w o f u r t h e r i n t e r s e c t i o n s coincide in a single c o n t a c t we impose t h r e e f u r t h e r condi- tions, t h u s r e d u c i n g t h e a g g r e g a t e of q u i n t i c curves to oo 5. C o n v e r s e l y : t h e r e are ~ 20 q u i n t i c curves a n d ~ 15 sets of t h r e e conics in a plane, so t h a t it would seem t h a t t h e r e is a finite n u m b e r of such sets of t h r e e conics associated w i t h a g e n e r a l p l a n e quiutic. A set of t h r e e conics which is such t h a t all t h e i r in- t e r s e c t i o n s lie on t h e quintic a n d each conic t o u c h e s the quintic m a k e s up a c o n t a c t sextic, i . e . a sextic which has t w o i n t e r s e c t i o n s with t h e q u i n t i c w h e r e v e r it meets it. I f F 1 ~ o , F 2 = o , F ~ : o are t h e e q u a t i o n s of the conics t h e e q u a - tion of t h e quintic can be w r i t t e n
t 1 1 ~ 1 -t- t 2 1 ~ -1 + t3 I~',~ 1 : O,
w h e r e t, ~ o , t,~ ~ o , t . ~ : o are t h e t a n g e n t s of t h e r e s p e c t i v e conics at t h e points w h e r e t h e quintic t o u c h e s t h e m . I f this could be established as a cano- nical f o r m f o r t h e t e r n a r y q u i n t i c t h e existence of the set of t h r e e conics w o u l d follow i m m e d i a t e l y .
S u p p o s e n o w t h a t we specialise t h e c o n f i g u r a t i o n . L e t one of t h e t h r e e conics b r e a k up into a line-pair and, as t h e conic t o u c h e d the quintic, let t h e i n t e r s e c t i o n of t h e two lines be on t h e quintic. Such a c o n f i g u r a t i o n of t w o conics a n d a line-pair t o g e t h e r c o n s t i t u t i n g a d e g e n e r a t e c o n t a c t - s e x t i c does n o t exist f o r a g e n e r a l plane quintic. B u t we shall see t h a t when we specialise t h e n e t of quadrics in [4] in a c e r t a i n way its J a c o b i a n curve is such t h a t a n y p l a n e quintic t h a t is in (I, I) c o r r e s p o n d e n c e with it has a d e g e n e r a t e c o n t a c t - s e x t i c of this kind; i n d e e d we shall a c t u a l l y o b t a i n t h e e q u a t i o n of such a plane q u i n t i c by e q u a t i n g to zero t h e d i s c r i m i n a n t of t h e special net of quadrics (w 5).
The plane configuration is specialised further when another of the conics becomes a line-pair whose intersection is on the quintic; we then have a plane quintic which passes through the six vertices of a quadrilateral and is such t h a t its remaining eight intersections with the sides of the quadrilateral are on a conic, this conic also touching the quintic. This type of plane quintic will also be obtained by specialising the net of quadrics in [4] and equating its discriminant to zero (w 12).
I f each of the conics is taken to be a line-pair whose intersection is on the quintic t h e n the quintic has an inscribed hexagram and is in birational correspondence with the Jacobian curve of a net of quadrics whose canonical form is II. The two special plane quintics alluded to above will thus arise from nets of quadrics in [4] intermediate between the general form I and the special form II.
The consideration of plane quintic curves has shown how we may expect to find nets of quadrics in [4] which are special, and yet not so special as the net I I ; similar consideration will also show how we may expect to find nets of quadrics in [41 t h a t are still more special t h a n II. The Jacobian curve of the net I I is such t h a t any plane quintic which is in birational correspondence with it has an inscribed hexagram; we then enquire as to how a plane quintic with an inscribed hexagram may be further specialised. One mode of specialisation is obvious; we may suppose t h a t the six sides of the hexagram, instead of being any six lines in the plane, are six tangents of a conic. I t is then found t h a t there is an infinity of hexagrams inscribed in the quintic curve, and t h a t the sides of these hexagrams all touch this same conic. The condition for the six sides of the hexagram to touch a conic can be written down at once in terms of the coefficients in I I ; it appears t h a t the Jacobian curve of a net I I which is specialised in this way has not twenty but an infinity of trisecants. The properties of this Jacobian curve and of loci t h a t are associated with it are obtained in w167 I5--49. This particular net of quadrics may well be regarded as the analogue, in [4], of the net of polar quadrics of points of a plane in regard to a cubic surface in [3].
We can also consider another method of'speeialising a plane quintie with an inscribed hexagram. We may always suppose t h a t the equation of such a quintie is
~-~ + ~-~ + ~-1 + ~-1 + ~;-~ + g~-~= o,
where the six lines ~ = o are the sides of the hexagram; then the t a n g e n t of
3 3 - - 3 5 1 5 0 . Acta mathematica. 66. I m p r i m 6 ]o 24 octobre 1935.
this curve at the vertex ~ = ~ j - ~ o of the hexagram is ~ + ~ j = o . Now the six linear forms ~ must satisfy three linearly independent linear identities, and we might enquire how the curve is specialised when we assume these linear identities to be of a special kind; it must always however be remembered that, as no three sides of the hexagram are to be concurrent, it must never be pos- sible to obtain any linear identity between less t h a n four of the forms ~. But let us suppose t h a t the identity
#, + ~ + ~ + ~, + ~ + ~ - o
holds. Then, if the six sides of the hexagram are divided into three pairs in any of the fifteen possible ways, each of the three pairs determines a vertex of the hexagram, and the three tangents of the quintic at these vertices are con- current; they are, indeed, concurrent in a point of the Curve. Thus, when this identity is satisfied, the fifteen tangents of the quintie at the vertices of the hexagram meet by threes in fifteen points of the curve.
We can also enquire whether there are quintic curves with inscribed hexa- grams such t h a t some but not all of the fifteen triads of tangents are concur- rent in this way. These and many other matters will be treated of in their proper place; enough has been said here to indicate some few of the results to which our investigations may lead.
The Jacobian Curve w i t h two P a i r s of C o n c u r r e n t Trisecants.
I. L e t us suppose t h a t through a point 0 of ~ there pass two different trisecants OPQ, ORS; call these t 1 and t~ respectively. Then the plane OPQRS is a secant plane of ~, and meets ~ in a sixth point Z; the trisecants t~ and t~ which are conjugate to ti and t~ both pass t h r o u g h the point I' of ~ to which this secant plane is conjugate, and the plane which contains them is the secant plane conjugate to O. Let t~ meet ~ again in U and V, and let t~ meet again in W and X.
The points Z, R, S are the three f u r t h e r intersections with ~ of a secant plane passing through the trisecant tl and conjugate to T; hence the five cones (Z), (R), (S), (U), (V), belong to a pencil; similarly (Z), (P), (Q), (W), (X) belong to a second pencil. But, if Z' is the sixth intersection of the secant plane T U V W X with ~, it follows, in precisely the same way, t h a t each of the two sets of five cones
(z'), (u), (v), (R), (s) and (Z'), (W), (X), (P), (q)
is a set of five cones b e l o n g i n g to a pencil. Thus t h e two cones (Z) a n d (Z') m u s t be the same, and the two p o i n t s Z a n d Z' m u s t coincide; t h e p o i n t of i n t e r s e c t i o n of the planes tit ~ a n d t~ t~ is on ~.
Since t h e f o u r cones (P), (Q), (W), (X) b e l o n g to a pencil of quadrics c o n t a i n i n g (Z) t h e solid P Q W X m u s t c o n t a i n t h a t secant p l a n e of ~ w h i c h is c o n j u g a t e to Z ; similarly this same s e c a n t p l a n e m u s t lie in t h e solid R S U V : t h e secant plane fl c o n j u g a t e to Z is t h e r e f o r e t h e p l a n e of i n t e r s e c t i o n of t h e two solids tit'2 and t'lt~; it m e e t s ,~ in 0 a n d T, and in f o u r f u r t h e r points E , F , G , H .
2. T h e p o l a r lines of a s e c a n t p l a n e of ~ in r e g a r d to t h e quadrics of t h e n e t g e n e r a t e a cubic cone whose v e r t e x is t h a t p o i n t of ~ to which t h e s e c a n t plane is c o n j u g a t e , a n d this cone contains t h e six s e c a n t planes of w h i c h pass t h r o u g h its v e r t e x (G.N.Q. w 9). I n p a r t i c u l a r t h e p o l a r lines of t h e p l a n e t~ t~ g e n e r a t e a cubic cone 1Io whose v e r t e x is 0; this cone c o n t a i n s t h e p l a n e fl a n d the plane t lt~; it also c o n t a i n s t h e two o t h e r secant planes, say 21 and t h , which pass t h r o u g h t I a n d t h e two o t h e r secant planes, say ~ a n d
#2, w h i c h pass t h r o u g h t2; tl a n d t~ are n o d a l lines on Ho. Since t h e plane v c o m m o n to the solids ~ltt I and ~tt,~ m e e t s 1Io in f o u r lines it m u s t lie entirely on 11o; m o r e o v e r v passes t h r o u g h t h e line OT since b o t h t h e solids ~1#1 a n d E2/t2 do so. T h e section of 11o by an a r b i t r a r y solid is a cubic surface H a w i t h two nodes N 1 and N~; it c o n t a i n s t h e line N1N2, two f u r t h e r lines 11 a n d m 1 t h r o u g h N1, two f u r t h e r lines 12 a n d m~ t h r o u g h N~, t h e line n in w h i c h t h e planes l~mt a n d l~m~ i n t e r s e c t a n d a line b w h i c h m e e t s n in a p o i n t B on t h e line OT. T h e p r o j e c t i o n of ~ f r o m 0 is a curve 0, of o r d e r 9 a n d g e n u s 6, on H'~; 0 has nodes at N1 a n d N 2 a n d m e e t s t h e line N 1 N 2 in one f u r t h e r point, also it m e e t s each of l~ a n d ml in t h r e e points o t h e r t h a n its n o d e N~ and e a c h of l~ and m s in t h r e e points o t h e r t h a n its n o d e N~, a n d it m e e t s b in five points, one of which is B. I t can be verified t h a t t h e c a n o n i c a l series is cut o u t on 0 by the cubic surfaces w h i c h c o n t a i n the t h r e e lines b, l~, m~ and w h i c h have a n o d e at N 1 a n d pass simply t h r o u g h N,,; the two nodes of 0 a n d its i n t e r s e c t i o n s with b,l~ a n d m~ are fixed points, n o n e of w h i c h b e l o n g s to a g e n e r a l c a n o n i c a l set. N o w a m o n g t h e s e cubic surfaces w h i c h cut o u t c a n o n i c a l sets on 0 t h e r e are t h o s e w h i c h consist of t h e plane llrn~ t a k e n t o g e t h e r with quadrics c o n t a i n i n g t h e line b a n d t h e two points N 1 a n d N2; t h u s t h e quadrics
cut o u t c a n o n i c a l s e t s on 0, b u t all t h e s e c a n o n i c a l sets include t h e p o i n t B since B lies n o t only o n each q u a d r i c b u t on the p l a n e
lira1.
Conversely: all t h e c a n o n i c a l sets on 0 to which B belongs can be o b t a i n e d as the i n t e r s e c t i o n s of 0 with quadrics c o n t a i n i n g the line b and the two points NI a n d N,, as such quadrics f o r m a system of f r e e d o m 4, one less t h a n t h e f r e e d o m of t h e c a n o n i c a l series on a curve of genus 6. H e n c e we m a y s t a t e t h e f o l l o w i n g :T h o s e c a n o n i c a l sets of ~ which c o n t a i n T are cut out by t h e q u a d r i c cones, v e r t e x 0, w h i c h c o n t a i n t h e plane ~ a n d t h e lines tl a n d t~; t h e five in- t e r s e c t i o n s of -~ w i t h #, o t h e r t h a n T, a n d the i n t e r s e c t i o n s of ~ w i t h t~ a n d t~, are n o t to be r e c k o n e d as points b e l o n g i n g t o t h e c a n o n i c a l sets. T h e t w e n t y i n t e r s e c t i o n s of such a q u a d r i c cone with ,,~ consist of 0, c o u n t e d twice, t h e e i g h t points P, Q, R, S, E , F , G, H a n d a c a n o n i c a l set of ten points, of w h i c h T is one; c o n v e r s e l y any c a n o n i c a l set to which T belongs can be o b t a i n e d in this way.
W e may, in particular, take t h e q u a d r i c cone to consist of t h e pair of solids t~ t~ a n d tl t~; t h e solid t~ t~ meets ~ in t h e set of points
(OPQT WXEFGH)
and t h e solid
t~to.
m e e t s ~ in the set of points(ORSTUVEFGH);
s u b t r a c t i n g f r o m t h e sum of these two sets t h e set(O~PQRSEFGH)
we find t h a t t h e set(TsUVWXEFGtt)
is a c a n o n i c a l set on #. Similarly, since t h e c a n o n i c a l sets of ,,~ w h i c h c o n t a i n 0 can be cut out by t h e q u a d r i c cones, v e r t e x T, w h i c h c o n t a i n t h e plane # and the lines t~ a n d t~, it is f o u n d t h a t t h e set(O~PQRS EFGH)
is a c a n o n i c a l set on ,% This l a t t e r r e s u l t also follows i m m e d i a t e l y f r o m t h e k n o w n r e s u l t(G.N.Q. w
28) t h a t , w h e n a q u a d r i c m e e t s ~ in all t h e points of a g i v e n c a n o n i c a l set, its r e s i d u a l i n t e r s e c t i o n s w i t h ~ also f o r m a c a n o n i c a l set; f o r we have j u s t seen above t h a t t h e set(O~PQRSEFGH)
is residual to all t h e c a n o n i c a l sets which c o n t a i n T.3. S u p p o s e n o w t h a t a b i r a t i o n a l c o r r e s p o n d e n c e is established b e t w e e n t h e c u r v e # a n d a p l a n e q u i n t i c r t h e n any c a n o n i c a l set on # m u s t c o r r e s p o n d to a canonical set on ~, i . e . to t e n points of ~ which lie on a conic. Also if five points of # are the vertices of five cones which b e l o n g to t h e same pencil, t h e five points of ~ which c o r r e s p o n d to t h e m m u s t be eollinear. H e n c e , if we d e n o t e c o r r e s p o n d i n g points of ~ a n d # by t h e same small a n d c a p i t a l l e t t e r , we have on ~ two sets of five collinear p o i n t s
(zuw~s)
a n d(zpqwx); a
conic t h r o u g h the points p, q, r, s a n d a conic t h r o u g h t h e p o i n t s u, v, w, x are such t h a t t h e i r f o u r i n t e r s e c t i o n s e , f , g, h are on ~, while the first of t h e s e conicstouches ~ a t a point o a n d t h e second touches ~ at a point t. A configuration of this k i n d does n o t exist for a general plane quintic curve; b u t it m u s t exist for a n y plane quintic which is in birational correspondence w i t h a Jacobian curve, of a net of quadrics in [4], t h a t has two pairs of c o n c u r r e n t trisecants.
4. L e t the equations of t h e three solids which contain the pairs of t h e three, secant planes t h r o u g h t~ be X x - ~ o, X~ = o, X 3 = o, t h e plane t~ t 2 being the plane of intersection of X ~ = o a n d X 3 = o ; let Y l - ~ o , Y 2 = o , Y . ~ = o be the equations of the solids which c o n t a i n the pairs of t h e three secant planes t h r o u g h t~, the plane t; t.~ being the plane of intersection of Y~ ~ o a n d Ya = o.
T h e n we m a y suppose t h a t a n y quadric of the n e t has an equation of the f o r m
where
a , X ~ + a2X~ + a3X~ + c Z ~ + b~Y[ + b.eY~ + b 3 Y ~ = o,
9 0 G.
This canonieal f o r m for the equations ensures t h a t tl a n d t~ are c o n j u g a t e triseeants of 9 ; in order t h a t t~ a n d
t~
should also be c o n j u g a t e trisecants t h e coefficients will have to be restricted in some way. This condition m u s t arise from the f a c t t h a t every point of t~ is c o n j u g a t e to every point of t~ in r e g a r d to every quadric of the net.The coordinates of 0 are (o, o, o; o, I, -- I); here we p u t the three X- coordinates before the semicolon a n d the three Y-eoordinates a f t e r it, the Z- coordinate being omitted. Suppose t h a t t 2 joins 0 to the point (aq, o, o; y~, y~, y~). Also let t~ join T, whose eoordinates are (o, I , - I; o, o, o) to t h e point (x~, x,'.,, x~; y~, o, o). A n y point on t2 has coordinates (x~, o, o; y~, y~ + Z, Y3 -- X)
p p ! P P
while a n y point on t2 has coordinates (xi, x2 + ~t, X a - ~t; yl, o, o); these two points are c o n j u g a t e if
t p !
aix~x~ + cxjy~ + b~y~y~ = o
o r
, + c +
yl xl ~ O .
The coefficients al, c, b 1 m u s t therefore be connected by this linear relation.
This relation m a y also be obtained by finding the condition t h a t the point of intersection of the two planes X~ = X3 = o a n d Y2 = Y~ ~ o should lie on 9.
The coordinates of this point are (I, o, o; I, o, o) and its polar prime w i t h re- spect tO t h e quadric is
a , X 1 + c Z ~- b 1 [I71 ~ o .
Since all these polar primes are t~o contain the same plane, whichever quadric of the net is taken, there must be a linear relation between the coefficients a,, c, b,. This linear relation can actually be found, because the plane t h r o u g h which the polar primes must pass is the plane, previously called {/, common to the solids t lt~ and t l t 2. The solid t~t~ joins the plane Z ~ X ~ - ~ o to the point
p p f f
(x,, x2, x~; y,, o, o), so t h a t its equation is
p p
y l X l - - X l Z = o .
Also the solid tlt2 joins the plane Z = Y~ = o to the poin~ (x~, o, o; yl, y~, y~), so t h a t its equation is
-- y~ Z + x l I71 = o.
We therefore have the relation
l a~, c,,
y l , - - X l ,
0, ~ Yl,
bl
0 ~ 0 , Xl
which is the same as that, previously found.
Conversely: suppose the quadrics of a net have equations of the canoni- cal form
alX~ + %X~ + a3X~ + c Z ~ + b l Y ~ + b~Y~+ b ~ Y ~ = o ;
then the two lines X I - = X ~ : X 3 : o and Y l = Y 2 = Y~--~o are conjugate trisecants of the Jacobian curve ~, three secant planes passing through each trisecant. L e t us also suppose, further, t h a t the coefficients of the three terms X~, Z ~, Y~ in the equation of every quadric of the net are subjected to a linear relation
c ~ Qa 1 + abl;
here of course Q and a are numerical constants, whereas the coefficients c, a,, bl differ for different quadrics of the net. Then, when this relation is satisfied, the two points
( I , O , O ; --0", I ~-~,, a--~,) and ( - - e , I +,u, Q - - # ; I , o , o )
are conjugate, in regard to every quadric of the net, whatsoever values Z and p
m a y have. W h e n ~ a n d tt v a r y t h e s e p o i n t s describe t w o lines w h i c h will be c o n j u g a t e t r i s e c a n t s of ~ ; t h e first of t h e m is a trisecant, o t h e r t h a n X , = X~ - -
= X3 = o, p a s s i n g t h r o u g h (o, o, o; o, I , - I) a n d t h e second is a t r i s e c a n t , o t h e r t h a n 171~- Y ~ = Y a = o , p a s s i n g t h r o u g h (o, ~, - - I; o , o , o ) . T h u s w h e n t h e coefficients c, a,, b~ s a t i s f y a l i n e a r r e l a t i o n t h e J a c o b i a n c u r v e h a s t w o p a i r s of i n t e r s e c t i n g t r i s e c a n t s .
W e c a n a l w a y s find a q u a d r i c b e l o n g i n g to t h e n e t in w h o s e e q u a t i o n t h e coefficients of a n y t w o of t h e seven s q u a r e s vanish. S u p p o s e t h e n we t a k e t h e q u a d r i c f o r w h i c h t h e coefficients of X~ a n d Z ~ b o t h v a n i s h ; t h e n , in v i r t u e of t h e l i n e a r r e l a t i o n c = @a, + abe, t h e coefficient of Y[ m u s t also v a n i s h . H e n c e t h e l e f t - h a n d side of t h e e q u a t i o u of this q u a d r i c is t h e s u m of o n l y f o u r squares, so t h a t t h e q u a d r i c is a cone w i t h v e r t e x X~ = X a - - Y= = Y~ - - o. H e n c e , when t h e r e is a l i n e a r r e l a t i o n c == @a, + abl, t h e i n t e r s e c t i o n of t h e t w o s e c a n t p l a n e s X ~ X a = o a n d Y ~ : Y ~ : o lies on ~.
5. T h e J a c o b i a n c u r v e & is in b i r a t i o n a l c o r r e s p o n d e n c e w i t h t h e q u i n t i c
~1~2~3(~2~]3 "~ g]3~l -~- VI~]2) ~- ~]1~]2~3(~2~3 -~- ~3~1 -~ ~I~2)
w h e r e ~, ~, ~ are l i n e a r f u n c t i o n s of t h r e e h o m o g e n e o u s c o o r d i n a t e s , a n d where, in a d d i t i o n , t h e r e is n o w a l i n e a r i d e n t i t y b e t w e e n ~ , ~, V,; say
w h e r e @ a n d a are n u m e r i c a l c o n s t a n t s . W e h a v e t h e i d e n t i t y
--(T@--lvI~2~3(V27~3 -~ ?]3Vl ~- ~IV'2 ~- G-'--I722I]3), so
that,
w r i t i n g~,,~3 + g3g, + ~1 ~,, + e - 1 ~,~.~ - - r ,
t h e e q u a t i o n to t h e quintic c u r v e is
( e L + a v , ) r d = eo--'~,7~v~F + o e - ~ 7 , G G A .
W r i t i n g , in t h i s e q u a t i o n ,
d - 7 , ( 7 d + 73)
~---1 72 78 = I q- O"
it b e c o m e s
,
e-~L~--
r - g, G + g~) I + @eG{(e + ,)~, + ( 0 + ~ ) v , } r n + ~,v,{e(e + ,)(w + w ) c + ~(~ + ~)(~.~ + ~ ) n } = o .
F r o m t h i s e q u a t i o n we can o b t a i n i m m e d i a t e l y t h e c o n f i g u r a t i o n of t h e p a i r of lines a n d t h e p a i r of conics, t h e e x i s t e n c e of w h i c h was e s t a b l i s h e d by t h e c o n s i d e r a t i o n of c e r t a i n c a n o n i e M sets on ~ ; t h e t w o lines are in f a c t ~l = o a n d 7~--~o, a n d t h e t w o conics are F~---o a n d d = o . T h e i n t e r s e c t i o n s of t h e eonic F = o w i t h t h e curve c o n s i s t of its f o u r i n t e r s e c t i o n s w i t h d - ~ o, its t w o i n t e r s e c t i o n s w i t h ~ = o, i t s t w o i n t e r s e c t i o n s w i t h 7, = o a n d its t w o inter- sections w i t h ~a + G = o; b u t t h e s e l a s t t w o i n t e r s e c t i o n s coincide in a single c o n t a c t , since G + G = o is t h e t a n g e n t of F - - - o at t h e p o i n t ~2 ~ G = o; t h i s s a m e line is also t h e t a n g e n t of t h e q u i n t i c a t t h e s a m e point. S i m i l a r l y f o r t h e conic J = o. F i n a l l y t h e i n t e r s e c t i o n of ~1 - o a n d 7~ ~ o is on t h e q u i n t i c curve.
T h a t a n y q u i n t i c c u r v e w i t h t h e s e p r o p e r t i e s is in b i r a t i o n a l c o r r e s p o n d e n c e w i t h a J a c o b i a n c u r v e ~ h a v i n g t w o p a i r s of c o n c u r r e n t t r i s e c a n t s follows a t once w h e n t h e l e f t - h a n d side of its e q u a t i o n is e x p r e s s e d as a s y m m e t r i c a l de- t e r m i n a n t of five rows a n d c o l u m n s (cf. G . N . Q . w 35).
6. W e h a v e seen h o w t h e g a c o b i a n c u r v e ~ a n d t h e p l a n e q u i n t i c ~ a r e specialised w h e n t h e coefficients in t h e c a n o n i c a l f o r m are s u b j e c t e d to a c e r t a i n t y p e of l i n e a r r e l a t i o n , a n d t h e q u e s t i o n n a t u r a l l y p r e s e n t s itself w h e t h e r f u r t h e r r e l a t i o n s of t h i s t y p e c a n i n t r o d u c e f u r t h e r speciMisations.
S u p p o s e we h a v e a p a i r of r e l a t i o n s s u c h as c ~ @a~ + ab i = ~a~ + wbe,
t h e coefficient a~ o c c u r r i n g twice, T h e r e is a q u a d r i c of t h e n e t in w h o s e equa- t i o n t h e coefficients of X~ a n d Z ~ b o t h v a n i s h a n d it follows, f r o m t h e e x i s t e n c e of t h e s e l i n e a r relations, t h a t t h e coefficients of :Y[ a n d Y~ also v a n i s h ; h e n c e t h e l e f t - h a n d side of t h e e q u a t i o n of t h i s q u a d r i c is t h e s u m of o n l y t h r e e squares, so t h a t t h e q u a d r i c is a c o n e w i t h a line f o r its v e r t e x . H e n c e , u n l e s s
t h e n e t of q u a d r i c s is to c o n t a i n a line-cone, t h e r e c a n n o t exist t w o such l i n e a r r e l a t i o n s ut t h e s a m e time.
W e m i g h t h o w e v e r h a v e a p a i r of r e l a t i o n s such as c : @~ a, + a, b~ = @., a2 + a.z b~,
in which n o n e of t h e six coefficients a~, a2, a~, b~, b,~, ba occurs twice. I n this case t h e e q u a t i o n of t h e p l a n e q u i n t i e is
~--1 + ~-1 + ~,~l + V~I +VV1 + ~]~-1 + ~(V~-I + ~}2 1 + ?}~-1)(~1 + ~-1 .~_ ~ 1 ) ~_ O, w h e r e ~ : o is t h e line w h i c h j o i n s t h e p o i n t ~ = ~]l = o to t h e p o i n t ~.~ : V.~ : o.
T h e J a c o b i a a c u r v e ~ n o w h a s t h r e e p a i r s of c o n j u g a t e t r i s e c a n t s t~, tl; t~, t~;
t3, t.~ w h e r e t~ a n d ta m e e t t 1 a n d t~ a n d t~ m e e t t~.
W e m a y also go a step f u r t h e r a n d s u p p o s e t h a t t h e coefficients in t h e c a n o n i c a l f o r m s a t i s f y t h e t h r e e r e l a t i o n s
c = @lal + (r~b L - - @.~a~ + a.eb~ = @.~a3 + o363.
A p l a n e q u i n t i c in b i r a t i o n a l c o r r e s p o n d e n c e w i t h t h e J a c o b i a n c u r v e of s u c h a n e t of q u a d r i c s is o b t a i n e d if we t a k e a t r i a n g l e f o r m e d by t h r e e lines ~ = o,
~.~ ~ o, ~3 ~ o a n d t h e n a second t r i a n g l e , in p e r s p e c t i v e w i t h this, f o r m e d by t h r e e lines ' h = o, ,;~ ~ o, ~a = o; we t h e n t a k e t h e q u i n t i c c u r v e
where n o w ~ = o is t h e axis of p e r s p e c t i v e . T h i s q u i n t i c c u r v e h a s t h r e e con- t a c t sextic curves e a c h c o n s i s t i n g of t w o lines a n d t w o conics. T h e c u r v e n o w h a s a p a i r of c o n j u g a t e t r i s e c a n t s t a n d t' such t h a t t h r o u g h a n y inter- section of t or t' w i t h ~ t h e r e passes a second t r i s e c a n t . T h e t r i s e c a n t s w h i c h pass t h r o u g h t h e t h r e e i n t e r s e c t i o n s of & a n d t are c o n j u g a t e to t h o s e w h i c h pass t h r o u g h t h e t h r e e i n t e r s e c t i o n s of ~ a n d t'.
The Jacobian Curve with four Concurrent Triseeants.
7. S u p p o s e t h a t a n e t of q u a d r i c s in [4] is s p e c i a l i s e d in s u c h a w a y t h a t t h e r e is one s e c a n t p l a n e a w h o s e six i n t e r s e c t i o n s w i t h t h e J a c o b i a n c u r v e a r e t h e v e r t i c e s of a q u a d r i l a t e r a l ; t h e f o u r sides tt, t~, t 3, t~ of t h e q u a d r i l a t e r a l are t r i s e c u n t s of ~. D e n o t e t h e p o i n t of i n t e r s e c t i o n of tl a n d t~ b y Pt~, a n d
34--35150. A c t a mathematica. 66. Imprlm6 le 24 octobre 1935.
similarly for the other five vertices of t h e quadrilateral. The four trisecants t'~, t'2, t'3, t'~ which are conjugate to tl, t2, ta, t,~ all pass through t h a t point 0 of 8 to which the secant plane a is conjugate and the six secant planes t h r o u g h 0, conjugate to the six vertices of the quadrilateral in a, are the six planes which contain the pairs of the four lines t'i t~, t~ ~, t'i.
Consider now the solid which is determined by the two secant planes t~t'~
and t~ ~t~, both passing through the trisecant t'~; since these are the secant planes conjugate to Pr) and P~a the solid must contain t h a t point of 8, which is the remaining intersection of & with t~, i.e. PLy, and P~,~ is the unique intersection of the solid with ~ which lies neither on the plane t~t~ nor on the plane t; t~.
I t therefore follows t h a t PI~ is t h a t intersection of the secant plane t~ t~ with,~
which lies neither on t~ nor on t[~. Hence the six points of ~ which lie in u are the intersections of a with the six secant planes through O; the secant plane which is conjugate to any vertex of the quadrilateral in a meets a in the op- posite vertex of the quadrilateral, and the three pairs of opposite vertices of the quadrilateral are pairs of conjugate points in regard to all the quadrics of the net.
8, The polar lines of a in regard to the quadrics of the net generate a cubic cone IIo, vertex 0, containing the six secant planes of # which pass through 0 and having" the four trisecants t'1, t,~, t~, t~ as nodal lines. An ar- bitrary solid, not passing through 0, meets H o in a four-nodal cubic surface and is projected from 0 into a curve 0, of order 9 and genus 6, lying on this surface and having a node at each node of the surface. The canonical series is cut out on this curve by the quadrics through its four nodes, and therefore the canonical series on # is cut out by those quadric cones with vertex 0 which coMain the four
! * t !
trisecants t~, t~, ta, t~, The twenty intersections of such a cone with ~ consist of O, counted twice, the remaining eight intersections of ,~ with its four trise- cants t h r o u g h O, and ten points forming a canonical set. In particular we may take the quadric cone to be a pair of solids through 0 and, if Oi, 0'~ are the two intersections, other than O, of # with the trisecant t~, we may note the following six canonical sets on &:
(02 o; o~ o~ ~'~ P2~ P.~ P~I P,2),
(On O; 01 O~ p2 ~, pa~, Pli P12 P2a),
(0101' 0202' P342 PH P2~ P2~ Pa),
(01 O; 04 O~ P~a Pa~ P~ P24 P34),
(0~ O~ O~ O~ V~, P , P~ V~, P~,),
(0~ O~ O~ O: P~ P~ P~, P,~ P~,).
T h e first of these, f o r example, is o b t a i n e d if the q u a d r i c cone consists of t h e
r r e t ! p
two solids tlt2t~ and t2t~t4; the o t h e r five sets are o b t a i n e d similarly. Also, since any q u a d r i c w h i c h c o n t a i n s a c a n o n i c a l set of points on 4t m e e t s ~ resid- ually in a n o t h e r c a n o n i c a l set, the p o i n t 0, c o u n t e d twice, a n d t h e e i g h t f u r t h e r i n t e r s e c t i o n s of ~ w i t h its f o u r t r i s e c a n t s t h r o u g h 0, m a k e up a c a n o n i c a l set on ~; h e n c e we m a y n o t e t h e s e v e n t h c a n o n i c a l set
(o ~ o~ o~ o,, o~ o~ o~ o~ o~).
9. S u p p o s e now t h a t t h e quadrics of t h e n e t are r e p r e s e n t e d by t h e p o i n t s of a plane, ~ being t h e r e b y p u t into b i r a t i o n a l c o r r e s p o n d e n c e w i t h a plane q u i n t i c curve ~; any set of t e n points of ~ w h i c h c o r r e s p o n d s to a c a n o n i c a l set on ~ m u s t consist of t h e i n t e r s e c t i o n s of ~ with some conic. N o w it is seen a t once, on r e f e r r i n g to t h e g r o u p of six c a n o n i c a l sets on ~, t h a t , so long as we do n o t take a pair of sets w r i t t e n in t h e same h o r i z o n t a l row, a n y two of these six sets have in c o m m o n six points of ~ ; the two c o r r e s p o n d i n g c a n o n i c a l sets on ~ t h e r e f o r e also h a v e six points in c o m m o n a n d so, since t w o different conics can only have m o r e t h a n f o u r c o m m o n points w h e n t h e y are b o t h line-pairs, with a line c o m m o n to b o t h pairs, it follows t h a t t h e six conics which cut out t h e six c a n o n i c a l sets on ~ are t h e six line-pairs w h i c h can be f o r m e d f r o m t h e sides of a q u a d r i l a t e r a l . Two c a n o n i c a l sets on ~ which c o r r e s p o n d to a p a i r of canonical sets of ~ w r i t t e n in t h e same h o r i z o n t a l row of t h e g r o u p of six are cut out by line pairs w i t h o u t a c o m m o n line; two such sets have f o u r points in common, and t h e s e points are f o u r of t h e vertices of the q u a d r i l a t e r a l ; t h u s t h e vertices of the q u a d r i l a t e r a l all lie on ~, b e i n g the points P~,P31, P~, Pt4, P~, P3, (we use, as h e r e t o f o r e , small l e t t e r s to d e n o t e p o i n t s of ~ which c o r r e s p o n d to points of ~ d e n o t e d by t h e c o r r e s p o n d i n g capital letters). I f we t a k e t h e t h r e e points p w i t h o u t the suffix i t h e n t h e set of five points w h i c h consists of t h e s e t h r e e points a n d the two points o~, o~ is a set of five collinear p o i n t s of ~ a n d is c o m m o n to t h r e e of t h e six c a n o n i c a l sets. I f we n o w r e f e r to t h e s e v e n t h c a n o n i c a l set m e n t i o n e d on ~ we see t h a t t h e e i g h t points o~, o~, o~, o~, %, 0~, o~, o~ lie on a conic, and t h a t t h e two r e m a i n i n g i n t e r s e c t i o n s of this conic w i t h ~ coincide in a c o n t a c t at o. W h e n c e we h a v e the f o l l o w i n g :
I f the Jacobian curve ~ of a net of quadrics in [4] is such that there is a plane which meets it iz~ the six vertices of a quadrilateral then, i f ~ is any plane quintic which is in birational correspondence with ~, these six
points of ~9 correspond to six points of ~ which are also the vertices of a quadrilateral. The remaining eight intersections of ~ with the sides of this quadrilateral are on a conic, and this conic touches ~.
I t should be noticed t h a t t h r e e c o l l i n e a r vertices of the quadrilateral on ~ cor- respond to three non-collinear vertices of the quadrilateral on ~, namely to the vertices of a triangle obtained by omitting one side of the quadrilateral. This configuration of four lines and a conic is clearly a special case of the configura- tion, obtained in w 3, of two lines and two conics; here one of the two conics touching the quintic curve has become a line-pair intersecting on the curve. We can also anticipate a further speeialisation of the present configuration, namely when both conics in the configuration of w 3 have become line-pairs intersecting on the curve; the quintic is then circumscribed to a hexagram.
Io. The existence of the quadrilateral inscribed in the plane quintic ~ can be established without any appeal to canonical sets. For it is known t h a t a secant plane which passes through a trisecant of ~ meets ~ f u r t h e r in three points which are the vertices of three cones belonging to the same pencil, the vertices of the two remaining cones of this pencil being those two points of ,~
which are on the conjugate trisecant and which are not conjugate to the secant plane. :Now the secant plane a contains the trisecant t4 and meets ~ f u r t h e r in the three points P ~ , P4,2, Pea; hence these three points are vertices of cones belonging to the same pencil. The vertices of the two remaining cones of this pencil are the two points, other than O, in which ,~ is met by the trisecant t~; i. e. they are the two points 04, 0~. Thus the five points 01, O~, P~4, P42, P,23 are the vertices of five cones of a pencil. Similar arguments give three other such sets Of five points, and it follows that, if ~ is any plane quintic in birational correspondence with ~, the four sets of points
(01 o'Pa~P4,2P~.~), (0`2 0"2pt, pl~p~4) (0,~ o~p12p,24Pa1) , (0 a O'~p,2.~p.~lpl,2)
are four sets of five collinear points. Thus the six points p~j are the six ver- tices of a quadrilateral inscribed in ~. I n order to establish the fact t h a t the eight remaining intersections of ~ with the sides of this quadrilateral are on a conic we show, as above, t h a t they correspond to eight points of ~ which be- long to the same canonical set.
One or two further features of the correspondence between ,~ and ~ may
be pointed out (cf. G.N.Q. w167 2I, 22). Since the six points Pfj lie in the secant plane of ~ conjugate to 0 there is a cubic curve touching ~ at each of the six points ply, the three remaining intersections of this cubic with ~ lying on the t a n g e n t of ~ at o. Again, the secant plane conjugate to P ~ meets ~ in the six points O, 01, 0'1, 0~, 0~, P ~ ; hence there is a cubic curve touching ~ at the six points o, 01, o'1, o2, o'2, p ~ and meeting ~ again in its three intersections with its t a n g e n t at p,~; there are five other cubic curves similarly associated with the five points pij other than Ply. Also, since the points 0, Of, O~ on are collinear there is a conic touching ~ at each of the points o, of, o~; thus we have three t r i t a n g e a t conics of ~ with a common point of contact. There is also a conic touching ~ at the vertices of any one of the four triangles formed by three of the sides of the quadrilateral, since the vertices of such a triangle correspond to three collinear points of ~. Since the two t r i t a n g e n t conics as- sociated with two conjugate trisecants of ~ are such t h a t their four intersec- tions lie on ~, the four intersections of the conic touching ~ at o, o~, o~. with the conic touching ~ at the three points p with the suffix i lie on ~.
II. We can obtain a canonical form for the net of quadrics when its Jacobian curve has a secant plane meeting it in the vertices of a quadrilateral and also, by taking the discriminant of a quadric of the net, obtain an equation for the plane quintic which corresponds birationally to the Jacobian curve.
We take the four solids
X I~0, X 2~0, X s : o , X 4~0,
to be those solids which pass through 0 and which contain the sets of three of the lines t'~, t'~, t'3, t'~, the solid X i ~ o containing the three lines other t h a n t~; also we take the plane a to be Xs---X 6 = o . The six forms X arc homo- geneous linear forms in five variables, so t h a t they satisfy one linear identity;
we may take this to be
X I + X 2 + X~ + X 4 + X s + X6---o.
The plane a contains four trisecants t 1, t~, t~, t, of the Jacobian curve, and the equations of the trisecant ti are Xf ~ - X 5 - X 6 = o. The four pairs of lines
have the property that, if one point is taken on each line of any pair, such a pair of points is eonjugate in regard to every quadrie of the neK I t then fol- lows easily t h a t ~he net of quadrics is given by a set of three equations of the form
a~ X~ + a~ X~ + a~ X ~ q- a 4 X ~ + a 5 X~ ~- a 6 X ~ q- 2 a ' X 5 X 6 = o l ]
X~ ~ ~ /
I l lbl X1 + b~
~ q- b 8 Xa -t- b 4 X~ + b 5 X~ + b 6 X~ + 2/~ X 5 X 6 ~ o,2 X 2 2
GX~ + c,~X~ + ca ~ + c ~ X ~ + c ~ X ~ + c ~ X ~ + 2 y X ~ X ~ - o .
This form I I I is not sufficiently general to represent a general net of quadries in [4], although it apparently contains the same number of constants as the general canonical form I; in fact we may, by using a suitable binary trans- f o r m a t i o n of X~ and X~, assume t h a t two of a, fl, 7 are zero.
I2. I f we write
~ x a i + yb~ + z c i , ~ ~ x a + y ~ + z 7 ,
the equation of an arbitrary quadrie of the net I I I is
~ x~ + ~. x~ + ~ x.~ + L x~ + ~ x~ + ~ x~ + 2 v x~ x~ o.
W r i t i n g ~his a s
~ ( G + x~ + x ~ + x~ + x,~) ~ + ~ x~ + ~ x ~ + ~ x~ + ~ x~ + ~ x~ + 2 ~ x~ x~ - o
we see t h a t the diseriminant of the quadrie is
D
The equation of the plane quintic is ]9 ~ o. W r i t i n g this in the form ( ~ + ~ d , + L~I~.. + ~ , ~ ) ( ~ - ~ ) + ~ (~,~ + ~ - 2 ~ ) - o,
we see t h a t the curve passes ~hrough the six vertices of the q u a d r i l a t e r a l f o r m e d by the f o u r lines
~ , = o , ~ = o , ~ = o , ~ , = o ,
and also t h r o u g h all t h e i n t e r s e c t i o n s of t h e s e f o u r lines w i t h the conic ~ s : 7 ~.
F u r t h e r m o r e the r e m a i n i n g two i n t e r s e c t i o n s of the conic w i t h t h e quintic are its two i n t e r s e c t i o n s w i t h t h e line ~.~ + ~ = 2 7 and, as this is a t a n g e n t of the conic, these two i n t e r s e c t i o n s coincide in a single contact. T h e t a n g e n t of at the p o i n t 5~ = _~j = o (i, j = I, z, 3, 4) is ~ + ~j = o. T h e p o i n t ~gij on ~ is t h e i n t e r s e c t i o n of t h e two sides of the q u a d r i l a t e r a l o t h e r t h a n ~s = o a n d
~ j = o .
This f o r m of t h e e q u a t i o n f o r ~ enables us easily to v e r i f y the existence of t h e c o n t a c t curves t h a t h a v e been m e n t i o n e d . F o r example, t h e cubic
~ . ~ + [~[~[, + [~g~g~, +
~ , ~
= ot o u c h e s ~ at e~ch o f the six vertices of the q u a d r i l a t e r a l f o r m e d by t h e f o u r lines ~: = o; its r e m a i n i n g t h r e e i n t e r s e c t i o n s with ~ ~re its i n t e r s e c t i o n s w i t h
~,~ + ~ = z T, a n d this is t h e t a n g e n t of ~ at 0. I f we write the e q u a t i o n of ~ as
~,~
( h ~ - 7 ~ ) ( ~ + ~,) +~ {(~ + ~ ) ( ~ -
7 ~) + ~ , ~ ( ~ + ~ - ~ 7)} = o ,we see t h a t the i n t e r s e c t i o n s of ~ with the cubic
(g, + ~ ) ( h h - 7 ~) + , ~ (h + g ~ - 2 7 ) = o
consist of t h e t h r e e i n t e r s e c t i o n s of ~ w i t h its t a n g e n t ~3 + ~, = o, o t h e r t h a n t h e p o i n t of c o n t a c t PJ2 of this t a n g e n t , and of t h e twelve i n t e r s e c t i o n s of the cubic with the q u a r t i c curve ~ 1 ~ 2 ( ~ 6 - 7 ~) = o. B u t it is clear, f r o m t h e f o r m . of its equation, t h a t t h e cubic passes t h r o u g h t h e i n t e r s e c t i o n of t h e two lines
~1 = o, ~ = o a n d also t h r o u g h t h e i n t e r s e c t i o n s of b o t h these lines w i t h t h e conic ~5~6=72; also t h a t t h e cubic t o u c h e s t h e conic at its p o i n t of c o n t a c t w i t h ~5 + ~ = 2 7; h e n c e t h e twelve i n t e r s e c t i o n s of t h e cubic a n d q u a r t i c coin- cide in pairs, and t h e cubic t h e r e f o r e t o u c h e s ~ at six points; these points are
p p
P34: 0: 01: 01: 02~ 02.
W h e n the e q u a t i o n of ~ is w r i t t e n
~ 3 ~ { h ~ - 7 '~ + ~ ( h + ~ - 2 7)} + ~ i ( h ~ - 7~)(~3~ + ~ + s ~ ) = o ,
we see t h a t the two conics
- v ~ + ~ (~., + i~ - 2 v ) = o ,
~.~, + ~!,, + ~ ~ = o,
are b o t h t r i t a n g e n t conics of ~ a n d t h a t t h e i r f o u r i n t e r s e c t i o n s lie on ~; t h e first conic t o u c h e s ~ at the t h r e e points 0, o~, 01 a n d t h e second conic t o u c h e s
at t h e t h r e e points P~2, P~a, P~a. A n d so on.
I3. T h e locus of p o i n t s in w h i c h a n y g i v e n s e c a n t plane a is m e t by t h e o t h e r s e c a n t planes of ~ is, in general, a c u r v e of o r d e r 13 with q u i n t u p l e points at t h e i n t e r s e c t i o n s of a w i t h # (cf. G.N.Q. w 6). I f h o w e v e r a m e e t s ,,~ in t h e six vertices of a q u a d r i l a t e r a l t h e sides of this q u a d r i l a t e r a l , each of which is a t r i s e c a n t of ~ a n d lies in two secant planes o t h e r t h a n a, all c o u n t twice as parts of the curve of o r d e r I3; h e n c e t h e r e s i d u a l p a r t is a q u i n t i c curve passing t h r o u g h t h e vertices of t h e q u a d r i l a t e r a l . T h u s , f o r this p a r t i c u l a r curve ~, t h e double surface of t h e locus R~ 5 g e n e r a t e d by the s e c a n t planes of contains a p l a n e quintic curve. This q u i n t i c curve circumscribes the quadri- l a t e r a l f o r m e d by t h e f o u r t r i s e c a n t s of # which lie in ~, a n d is in b i r a t i o n a l c o r r e s p o n d e n c e w i t h ~.
A p a r t f r o m t h e f o u r t r i s e c a n t s of # which lie in a a n d t h e f o u r t r i s e c a n t s of # w h i c h pass t h r o u g h 0 ~ has twelve o t h e r t r i s e c a n t s , these c o n s i s t i n g of six c o n j u g a t e pairs.
I4.
f o r m is
w h e r e
T h e J a c o b i a n C u r v e w h o s e T r i s e c a n t s a r e t h e E d g e s o f a H e x a h e d r o n . W e p r o c e e d now to c o n s i d e r the n e t of quadrics whose c a n o n i c a l
2 . X 2 2 ]
a I X L + a , X ~ + a~ 3 + a a X ~ + a , ~ X s + a 6 X ~ = o ,
~ + X 2 ~ 2
J
b i X ~ + b~X~ + b.X~ b4 4 + b ~ X s + b 6 X 6 = o ,
2 X 2 2
C 1 X ~ J[- C 2 X 2 JV C 3 3 -]- C4 X 2 JF C 5 X 5 + C 6 X 6 2 ~ O,
I I
X L + X 2 + X a + X 4 + X,~ + X G ~ o .
T h e six solids X = o are t h e faces of a h e x a h e d r o n ; this h e x a h e d r o n , w h i c h we m a y call ~, has fifteen vertices, t w e n t y edges, a n d fifteen planes. Since we can combine the t h r e e e q u a t i o n s I [ l i n e a r l y so t h a t a n y two of t h e six squares
disappear it follows t h a t each of the fifteen vertices of ~ is the vertex of a cone belonging to ~he net; hence the Jaeobian curve ~ is circumscribed to ~. Moreover the polar prime of any vertex of ~ in regard to every quadric I I contains the opposite plane of ~ (the plane of intersection of two faces of ~ being 'opposite' to that vertex which is the intersection of the remaining" four faces), so t h a t the fifteen planes of ~ are secant planes of 4t, being conjugate, respectively, to the opposite vertices of 9. Also, since each edge of ,~ contains three vertices, the twenty edges of ~ are trisecants of O. and each plane of f2, us it contains four edges, meets ,,~ in the six vertices of a quadrilateral. The twenty trise- cants of ~ are thus all accounted for in this way; the trisecants of the Jacobian curve of a net of quadries given by the canonical form I I are the edges of a hexa- hedron. Two trisecants are conjugate when they are opposite edges of gJ; i.e.
when the three faces of ~ passing through one of the trisecants and the three faces of ~ passing through the other together constitute all the six faces of &.
The double surface of R~ 5 now contains fifteen plane quintic curves; these are all in birational correspondence with ,,a and with each other.
Any solid which contains a secant plane of ~ meets ~ f u r t h e r in four points which are vertices of four cones belonging to a pencil; hence, if ~ is any plane quintic in birational correspondence with &, the vertices of the tetrahedron in which any face of ,~ is met by four ~)f the other faces correspond to four collinear points on ~. I t follows immediately t h a t any five faces of ~ form a simplex, inscribed in &, whose five. vertices correspond to five eollinear points of ~, and hence t h a t the fifteen vertices of ~ correspond to fifteen points of which are the intersections of six lines. Hence
I f the Jacobian curve of a net of quadrics in [4] is circumscribed to a hexahedron, then any plane quintic which is in biratio~al eorresponden.ce with it is circumscribed to a hexagram.
The intersections of the quintic with a side of the hexagram correspond to tile vertices of a simplex formed by five faces of the hexahedron; the ten ver- tices of a pentagram formed by five lines of the hexagram correspond to the ten intersections of the Jacobian curve with a face of the hexahedron. The ten vertices of the pentagram must therefore be the points of contact of the quintic with a contact quartic. Also the vertices of the quadrilateral formed by any f o u r lines of the hexagram correspond to the six intersections of the Jacobian curve with a plane of the hexahedron; hence there is a cubic curve touching
35--35t50. A c t a mathematica. 66. l m p r i m 6 le 24 o c t o b r e 1935
the quintic at each vertex of the quadrilateral, the remaining three intersections of the curves being the intersections of the quintie with its t a n g e n t at the point of intersection of the remaining two sides of the hexagram. T h e vertices of a triangle formed by three sides of the hexagram correspond to three collinear points of the Jacobian curve, so t h a t there is a conic touching the quintic at the vertices of such a triangle. I f we take two such triano'les whose sides con- sist of all the six lines of the hexagram the associated triseeants of ~ are con- jugate, so t h a t the four intersections of the two t r i t a n g e n t conies are on the quintie curve.
The equation of the quintie is obtained immediately by writing" zero in- stead of ~ in the d e t e r m i n a n t D of w I2; it is t h e r e f o r e
and the curve passes t h r o u g h the intersection of any
~ i = o. The quartic curve
~-1 .A V ~-1 _~ ~-1 _~_ ~---1 JV ~-1 = 0
two of the six lines
touches the quintie at the ten vertices of the p e n t a g r a m whose sides are tile sides of the hexagram with ~ - - o omitted; also the equations of the cubic cur- ves and t r i t a n g e n t conics just referr6d to can at once be w r i t t e n down.
Although every plane quintic which is in birational correspondence with a 3aeobian curve having an inscribed h e x a h e d r o n has an inscribed hexagram it is not true, conversely, t h a t every Jacobian curve which is in birational cor- respondence with a plane quintie having an inscribed h e x a g r a m has an inscribed hexahedron. Nevertheless, when a plane quintic has an inscribed hexagram, a Jacobian curve can always be found which is in birational correspondence with it and which has an inscribed hexahedron. For we cau always suppose t h a t the equation of such a plane quintic is
where
~i ~ x a ~ + y b i + z c i = o ( i = I , 2 , 3 , 4 , 5 , 6 ) are the equations to the sides of the hexagram. The equation to this quintic, when written in the determinantal form, is at once seen to be the condition t h a t a quadric of the n e t [ I should be a cone, mud the Jacobi~n curve of t h e net I I fulfills the required conditions.
I5. The locus of the poles of any solid S~ in regard to the quadrics of the net is a sextic surface on which lie ten lines. The line conjugate to any point of this surface lies in S 3 and, conversely, if a point is such t h a t the line conjugate to it lies in S~ then the point must lie on the surface
(G.N.Q. w
I3).Consider now the particular case when $3 is a face of the hexahedron. The remaining five faces of the hexahedron form a simplex whose ten edges are trisecants of &; the line conjugate to any point of any of these trisecants is the conjugate trisecant of ,a, and lies in $3. Hence the sextic surface which is the locus of poles of S~ passes t h r o u g h the ten edges of the simplex, and these are the ten lines which lie on the surface.
T h e J a c o b i a n Curve w i t h a S c r o l l o f T r i s e c a n t s .
I6. W h e n a plane qnintie has an inscribed h e x a g r a m the sides of the h e x a g r a m will not, in general, touch a conic. W e now prove t h a t if a plane quintie has an inscribed h e x a g r a m whose sides touch a conic then it has an infinity of inscribed hexagrams whose sides all touch this same conic.
L e t us suppose t h a t the hexagram consists of the tangents to the conic
xz-~y~ at
the six points(0~, 0i, I),
with i = 1,2, 3, 4, 5, 6. Then the equation of the quintie is of the form_ Voi + - - ~ i = l
Now take any point on this curve; if ~ and ~p are the parameters of the points of contact of the two tangents which can be drawn to the conic from this point,
E _ - 0 , ) - o
I f 99 is givell this is a quintic equation for ~p; it gives the parameters of the points of contact with the conic of those of its t a n g e n t s which pass t h r o u g h the five intersections of the quintic with the t a n g e n t to the conic at the p o i n t whose p a r a m e t e r is 9~. Suppose t h a t ~Pl and ~p~ are two roots of this quintic.
Then
Zi (~9 -- 0/) - I (~1 - - 0i) -1 = o, ' ~ '~t ((~ --- 0;) - 1 (~f12 - - Of) - 1 - - - O .
