393

CREMONA TRANSFORMATIONS IN

GEOMETRY OF MATRICES

by L. S, 6oddard (Aberdeen)

THE

I N T R O D U C T I O N

1. Let X and Y be square matrices of order n, whose elements are initially indeterminate over a base field F, which assumed to be wi thout characteristic and algebraically closed. We shall be interested only in the relative ratios of the elements of a matrix, so that X and ~ X are to be regarded as identical for all

E F (x~), where F (x,j) is the field obtained by adjoining to F the n ~ elements x,~ of X.

A simple and well-known identity in the theory of matrices is

aaj (acq x ) -- x i x I '~-2,

where adj X denotes the adjoint of X. We consider two matrix transformations, T and T*, based on this identity. These, together with their inverses are

T : Y ' - a d j X , T - t : a X - - adj Y,

where ~ = I X i"-2; and

T*;~ --IXE + X[, Y = ( k E - - X ) a d j (;~E + X),

T*- ' : ~ k --- ] vt E -r Y [, p X - - (~ E - - Y) adj (tL E + Y),

where E denotes the unit matrix, ~. and It are indeterminate over F, and - - g , , (k p.),-t. In this paper T and T* are considered as Cremona transfor-

mations between two projective spaces, and our main aim is to characterize the fundamental points and their transforms. The transformation T has appeared

394 L. s. CODDARD

several times in the literature, for the cases n --- 3 and n - - 4. But here we study T for general values of n, and pay attention to the special results holding when X and Yare specialised to be either both symmetric or both skew.

A l e m m a o n t h e a d j o i n t m a t r i x

2 . - - When the matrix X is specialised to a matrix Xo, whose rank is less than n - 1, adj Xo is null. In this case, to find the matrices Y corresponding

to Xo, it is necessary to examine the ~ adjoint of the ne ighbourhood, of Xo,

that is, adj (Xo § ~ Z), where z is a small parameter, with ultimately �9 ~-O, and Z is a matrix of indeterminates. The result is contained in this

L e m m a : Let Xo be a square matrix of order n and rank n - - r, and Z a matrix o f indeterminates. Expand adj (Xo § ~ "~) into a polynominal in ~. Then the coefficient of the lowest power of r is a matrix of rank r. Proof: Since Xo is of rank n - - r, all minors of Xo of order greater than n - - r vanish, and, in the expansion of adj (Xo § ~ E), the constant term and the

coefficients of e, ~ , . . . , ~-~ all vanish. We have

Y = add <Xo + ~ Z) == ~'-~ (,~ + ~ ,~0,

where ,~ and A, but not ~ and A' are independent of ~. ,~ is clearly a function of

the elements of Xo and E. N o w consider adj C~ (Xo + ~ Z), the k '~ adjugate compound of Xo § ~ Z, whose elements are the minors of Xo § ~ E of order n - - k. Such a minor, if k ~ r, begins with a term independent of e, and we have

adj(~) (Xo + ~ a) = ~,: + ~ ~

where ~., - - adj {~) Xo, and ~ depends on ~. Now take the k *h compound of Y

and use the result,

{adj (Xo -Jr- ~ _r)}{~) := IX o § e ~ [~-~ adj{ ,~} (Xo § ~ "~).

We obtain

y<~l = ~{~-,) (~ § ~ ~ ) l ~) - - I Xo § ~ "~ I ~-~ adj {~) (Xo § ~ 8,)

= ~c~ ~ {a § ~ a') k-I (y,~ § ~ ~,~).

CREMONA TRANSFORMATIONS IN THE GEOMETRY OF MATRICES 3 9 5

Hence

The left side begins with the constant term ~/,,) and the right side with ~- ' . Hence

,~<~) ~ 0 (k > r),

,~/*!-- A,-I ~ (k - - r).

Since Xo is of rank n - - r it follows that ,~ ~ adff) Xo ~ O, and hence ~ is of rank r.

T h e t r a n s f o r m a t i o n T

3 . - - Let the elements of X be coordinates of the generic point of an [n ~ - - 1], Y~, and, similarly, let Y be the generic point of an [n ~ - - 1], ~2. Then T is a Cremona transformation between ~2~ and x2~ by a system of primals of order n - 1. Also T is symmetric. In fact, if we identify ~2~ and Y'2, T is invo- lutory, that is T ~ = 1. in view of this symmetry it is only necessary to examine the fundamental points of T and the transformation of their ne ighbourhoods; the results for T -~ will be the same.

When X is specialised to a generic symmetric matrix, adj X is al- so symmetric, and T becomes a transformation, say 7"t, between two

~ (n - t - 2 ) ( n - l)['s.�9 Similarly, if X is of even order and is specialised to

the generic skew matrix, ad] X is also skew, and T becomes a transformation,

say T2, between two I ~- (n - - 2) (n - . l)J ' s. These transformations, T~ and T~

will be examined separately after the general case. The fundamental locus of T is given by adj X - - O, that is, by the vani-

shing of all minors of order n - - 1 in X. It is therefore the determinanial key- locus K~ (''-2' [(1), chapter V], which is of dimension n ~ - 5 and order

1 n e (n ~ -- l). On Ir (''-2/, there is a chain of multiple loci,

12

KI (~ C KI (2) C . . . C K, ('-~1

the points of KI It) which are not on Kt r being represented by the matrices of rank r.(t)

396 r.. s. ~ O . D A R D

The transforms of the fundamental points of T have been studied when n - - 3 and n --- 4, by Rozet (2) (3), who regarded X as defining a correlation betweea two [n - - l]'s. His method, based on the theory of singular correlations

may be developed into one applicable for all values of n (k); but a shorter and more powerful method is to use the above Lemma in a direct algebraic approach. Let X be a point on K~ (~) but not on /(t (~-~), so that X is of rank r. The point X + :--', in the neighbourhood of X is transformed, according to the Lemma,

into the point

where ,~. is of rank n - 1, that is ,~ is a point on K~ ('~-'). Hence a point in the neighbourhood of a given point on K~ (~) is transformed by T into a point on K~ I~-'). By varying ~" we obtain, on K2 ('~-~), the locus L which corre- sponds to the neighbourhood of X. We show that L is a linear space and that there is on /(t I~) a unique linear space through X such that to each point of this space there corresponds the same space L. It is known that K/~/ consists

of a single system of oo ~/'~-'/ [r ~ - l]'s, t !~! (I , p. 100). in terms of these

spaces the results may be stated in

T h e o r e m 1: To a point X on Kt I,), but not on K~ (~-l), there corresponds, on K2 (~-') an [(n .... I) ~ - - 1], t2 i~-'). Moreover, to every point of the unique [r = - - 1], tt It), t hou h X (which is not a point of K~(r-i)), there corresponds the same t~ ("-'). Proof: We consider the transformation of the typical t~('~, given by (")

o)

where P is a non-singular matrix of order r,

in tt ('). Let

: a d j ( X + ~ ) = a d j ( P Y

whose elements are coordinates

w ~ ~ " 2

E --3, ~ ~ 4 /

where the matrix -~ is partitioned in the same way as X, so that 5 4 is of order

n - - r. In the expansion of Y, in powers of r the coefficients of 1, ~, e ~ , . . . , t . . . . . ~ all vanish and we have

C R E M O N A T R A N S F O R M A T I O N S i N T H E GEOMETRY OF M A T R I C E S 397

where ~ is the matrix whose (i, j) th. element is the coefficient of ~ . . . . ~ in the expansion of the co-factor of the (j, /) th. element of X -]- ~ ~. By considering an element in any of the first r rows and first r columns of X + ~ E, it is clear that the co-factor of this element commences with a term involving r Hence ~ is of the form o) and it is easy to show that ~4 ---- [ P t ad j -4. Suppressing the factor e"-"-~ I P I it follows that a point of ti I'l is transformed into the point

y__(0 0) 0 L

where }'4 : adj ~4, and Y is independent of P, -vl, E~ and -'-3. It is now clear

that, for varying -", Y generates an [ ( n - - r ) 2 - 1],te C .... )onK.~" ..... ) .For ( O 0 / \ U ~, / 4

is the generic point of an [(n - - r) ~ - - i], t~ ~ ... . I on K1 ~ .... ~ and the t,,c"-rJgene- rated by Y is the transform of tl "'-t) by a Cremona transformation of precisely the type we are considering. Hence to the point

there corresponds the t,~ ~'-'1 defined by

for arbitrary Q. Moreover, since P does not appear in Y (it is removed in the factor I P I) it follows that the same t.J "-~) corresponds to every point of he t~ It) defined by X, provided this point does not lie on K '.r-'. This completes the proof of the theorem.

4. - - E x a m p l e s : (i) n = 3. T is a transformation between two [8]'s by quadric primals through a Segre Variety, V 2 , 2 - - K 1 ll). Details are given by Rozet (2).

( i i ) n = 4. T is a transformation between two [15]'s by cubic primais through K1/~l, which is of dimension 11 and order 20. This case is also studied 26 - R e n d . C i rc . M a t e r n . P a l e r m o , - - s e r i e I I - t o m o IT - a n n o 1953

3 9 8 L. s . O O V D A R ~

by Rozet (3), w h o determines the orders (and intersection properties) of the va-

rieties which correspond to linear subspaces.

(iii) general value o f n: write v - - - n 2 - 1. cons ider the order t~ of the

variety V~ which cor responds to a [~] of [v] under T. This order is equal to

the number of points common to V~ and a generic [v - - ~]. Owing to the sym-

metry of T this is equal to the number of points c o m m o n to Vv_~ and a generic [~],

which is t ,_~. Thus t~ = t~. where ~ -~- ~" = v.

The first few orders tv_~, t , _ 2 , . . . may be easily calculated. W e note that the base

locus K/'~-2~ is of dimension n ~ - - 5 and order 1 (n ~ _ 1). Thus t~_~, t,_~ 12

and tv_~ are equal to n - - 1, (n - - 1) 2 and (n - - 1) 3, respectively. The variety

V,_3, of order (n - - 1) 3, which cor responds to a Iv - - 3], is the complete in-

tersection of three primals ~ , ~2, r of the sys tem defining T. It therefore

contains K 1'~-2/, w h i c h is double on the primal K = [ X I : 0, and hence cuts

K in a variety of dimension v - - 4 which reduces to K !'-~t and a variable va-

riety, w h o s e order r is given by

r ----- n (n - - 1 ) 3 - - 2 �9 1 n 2 (n ~ 1) 1 12 (i

i2 (n - - 1) (n - - 2 ) ( 5 n - - 3).

The variety V~_4, of order t~_,, wh ich cor responds to a [v - - 4], is the

residual intersection of four primals k b ~ , . . . , r These all contain K I~-~l,

which is also of dimension v - - 4; so that

and

(co I . . . . . r = K <~-~i + V~_,,

1 tv_, : (n - - 1) 4 - - - - n ~ (n ~ - - 1)

12

1 == _= (n - - 1) (n - - 2) (11 n 2 - - 15 n -[- 6).

12

It is not poss ib le to proceed further than this because of our incomplete

knowledge of the enumerative characters of determinantal loci. The next problem

CREMoNA TRANSFORMATIONS iN THE GEO.METRY OF MATRICES 3 ~

is to determine the order, i, of the intersection U, of K ("-~) and V~,_,. Take a

section by a generic [5]. Then we obtain four primals, of order n - - 1, intersecting

in two curves, C,, of order _!_ n e (n 2 _ l) and C~, of order tv_ ,. Also i is the 12

number of points of intersection of C~ and C 2. If ~ is the rank of C~ we have (5),

1 n2 (n ~ 1) [4 (n 1) 4] i ~- e - : . . . . 12

n ~ (n 2 - - l ) ( n - - 2),

so the problem is reduced to this: what is the rank of the curve which is the

section of K/'-~) by a generic [5]? If i were known the order of the variety (of

dimension v -- 5 ) which is the variable intersection of V~_ 4 and K would follow,

for this is n tv_4 - - 2 i (the factor 2 since K (*'-~) is double on K). Also, the order

Iv-5 of the variety corresponding to Iv - - 5] would follow. For a primal cl~s cuts

Vv_~in Vv_ s and U, so that. tv_ , ~ - i - - (n -.- I ) t ,_,.

T h e t r a n s f o r m a t i o n , 7"t, d e f i n e d b y a s y m m e t r i c m a t r i x

F1 7 5 . - - TI is a transformation between two [2 n ( n - 1 ) ( n ~ 2)['s, X' 1

L J

and ~'~. The fundamental locus in ~,'~ is YI ('-2) [(1), p. 133], on which there is a chain of multiple loci

YI (1) C }'~J C . . . C }'1 ~'~-~)

In terms of the generators, u v), of Y('), we can, by an application of the Lemma deduce

V (,-r) T h e o r e m 2: To a point X on Y~(~) there corresponds, on - , . , a

[12 (n -- r) (n - - r -~- 1) - - 1], u2 (" -~).Moreover, to every point of the unique

[-~ r (r ~-, 1 ) - - 1],ui( ' ) , through X, which is not a point of Y,(~-~), there cot -

responds the same u~ I"-').

400 L. s. GODDARn

The proof of this theorem is similar to that of theorem 1.

Examples: (i) n : 3. 7"t is the familiar transformation between two [5]% by quadrics through a Veronese surface;

(ii) n --- 4. This case has been studied by Rozet (6). 7"t is a transformation between two [9]% by cubic primals through a V8 ~~ y/~).

The trans format ion T~ defined by a skew m a t r i x S of even order

6. - - if S i s a skew matrix of order 2r it is easy to show that adj S is skew; and it is well known that I S I - - where the Pfaffian A is of order r in the elements of S. When A is zero, S is of rank 2 r - 2, since a skew matrix is necessarily of even rank. Hence adj S =---O, when A : 0, and it follows that A is a factor of every element of adj S. We write

ad/ S : A S+,

where S + is a skew matrix whose elements are polynomials of order r - - 1 in

the elements of S. The matrix S+ will be used below to define a Cremona transformation.

The elements of S + are obtained as follows. Denote by Aij the Pfaffian whose square is equal to the minor of order 2 r - - 2 obtained from S by omit-

ting the i th. and j th. rows and the i th. and j th. columns. The (i, j) th. ele- ment of adj S is a minor of S, of order 2 r - - 1, which includes 2 r - - 2 ele-

ments of the principal diagonal, namely all elements except those in the i lh. and j th. places. W e find that

(adj S)o - - A ,X,~

so that the elements of S + consist o f the r (2 r - 1) Pfaffians signs) derived from the principal minors of S o f order 2 r -- 2.

[t is to be noted that, corresponding to the identities,

(with suitable

S ady S = r., I ad j S I = ~-"''"-",

there are the identies

S S+ - - S~ S -= a E, [ S+[ - - a~'"-".

CREMONA TRANSFORMATIONS I N T H E GEOMETRY OF MATRICES 401

The second of these follows from the fact that

l adj S I - - A2r I S+ I.

It is next observed that

and also

adj (adj S) = A,(~,'-~) S,

adj (aclj S) - - adj (A S +) - - A ~-~ adj S+ - - A ~ - , A ~-, S ,~ +

Hence S + + : At-2 S.

7. - - This identity allows a Cremona transformation to be set up as follows. Let S and T be generic skew matrices, each of order 2r, and take the elements of S and T to be the coordinates in a pair of [(2 r + 1) (r - - 1)]'s, ~1" and ~'2", respectively. A Cremona transformation, 7"2, between ~.," and }2(' is defined by

7"2 : T - - S +, T~ - t : a S - - - T +,

where a 2 -= I S I r-2. A point So on the fundamental locus satisfies So+ - - 0, which means that all the principal minors of order 2 r - - 2 in So vanish. Hence all the minors of this order vanish and So is of rank at most 2 r - - 4. The fundamental locus in }3/' is therefore the key-locus ~(2~-4) and 7'2 is a transform- ation by primals o f order r - - 1 through fl(2r-41.

Contained in fl(~-4) is the chain of multiple loci,

Also [(1), p. 200] Q(') is the locus of 00 *(2~-') [ 1 [2

transforms of points of the fundamental locus generators. We have

3 t ( t - - 1 ) - l[ 's , #'). The

.I

are given in terms of these

T h e o r e m 3: A point S on s is transformed into a

[ ( r - - t) (2 r - - 2 t - - 1) - - 1], w2 (2~-2') on O2(2r-~0.

Moreover, every point o f the unique [t (2 t - - 1) - - 1], w,/20 defined by S, which

402 L. s. CODDAaD

is not a point o f .Q <~,-2) is transformed into the same w~ I~'-~').

Proof: Consider the w~ 12') whose generic point is

s :

where S~ is a skew matrix of order 2t, whose elements are all indeterminate S~ is then non-singular. If (3 is an arbitrary skew matrix we have to consider M + where

M - s + , (3 = (s , + , (3. ~ (3~, (33, ~ (34/

the partitioning of (3 used here being obvious. We are interested in the matrix coefficient of the lowest power of ~ in the expans ion of M+ as a polynomial in t. We recall that the squares of the elements of M + are the principal minors of M of order 2 r - - 2. An examination of the possibilities shows that the lowest power of t in a principal minor of order 2 r - - 2 is ,2,-2,-2 and that the only principal minors of this order which possess a non-zero coefficient for ,~,-2,-~ are those involving the first 2r rows and columns. Such a minor is of the form

0 K I S i 0J __-- $t [ K ,

where K is a principal minor of (34 of order 2 r - - 2 t - - 2. Write A~ = S~. Then, omitting the factor ~2,-2~-2, it follows that

(00) m + : ( S + , ( 3 ) + = ' 0 (34+"

/ q N\ Hence, to the p o i n t S - - ( O 0 ) o f #2,), there corresponds the w, (v-2.) defined

l (:0! by A = (34 + �9 Moreover, since the elements of Si do not appear in A (they are

removed in the factor At) it follows that the same w~ (~'-2') corresponds to every point of the wi (~') defined by 8, provided ] $t j -# 0, that is, the point does not lie on f~(2,---~). This completes the proof of the theorem.

We now consider the transformation of a space #~,+tl. For this purpose we

consider the #2,+t), given by ($t O~ where Si is of order 2 t - } - 1 . The dimen- \ u U/

CREMONA TRANSFORMATIONS 1N THE GEOMETRY OF MATRICES 4 0 3

sion of the system of spaces w (~') in w I~'t~) is 2t, since these spaces are in (1,1) correspondence with the primes of [2t], and a set of 2 t -~- 1 linearly indepen- dent spaces w/'z'l defining the system is given by omitting successively in St the i th. row and column, for i - - 1, 2 , . . . , 2 t -]- 1. These spaces w/2`) are tran- (0 ~ 0) sformed into spaces w/2"-~'/ all passing through the w~ (~'-~'-~/ given by S~

where S~ is an arbitrary skew matrix of order 2 r - - 2 t - - 1. Thus the system of spaces w (~'~ contained in w/2'+ l) is transformed into the system of spaces wz <~'-~'/ passing through w2 (2~-~-~/" But these are the generators of a cone (1, p. 205), and we have

T h e o r e m 4 : A space wi I~-'§ is t r a n s f o r m e d into a cone

17<~"-~"-~ - - ( I 2 r - - 2 i - - 1, 2 t ,--[- 1 !,, [(r + t) (2 r - - 2 t - - 1) - - 11)

in such a w a y tha t spaces w (~'~ con ta ined in w/e'i tl are t r a n s f o r m e d into the

genera tor s o f this cone.

$. - - E x a m p l e s : (i) n - - 4. T~ is a collineation between two [5]'s.

(it) n - - 6. ?'2 is a transformation between two [14]'s, x2~" and ~z" by a system of quadrics tlarough firs), whish is a Vs 14, the Orassmannian of lines of [5]. To the points w/~) of or2) there correspond the [5]'s, w <~!, on 0~4) and to a plane

w ~s) of fltp there corresponds a cone

= (] 3, 3 i,, [ I I ] )

which is of order 6 and dimension 7. We now determine the orders and intersection properties of the varieties which

correspond to linear subspaces of 2i" . Firstly we notice that the base is of di-

mension 8, so that we have the following:

[13]->- V:,, [12]-~ V:,, [ll]->- V~. [lO]->- V:'o, [9]->- V;'.

The varieties V are the complete intersection of 1, 2, . . . , 5 quadrics. Now 6 quadrics cut in a V~' which contains ~/~), so the residual intersection is a V~ ~ Suppose this variety cuts .q~) in a I/7 '~, of order m. To determine m take a sec- tion by generic [7]. We obtain 6 quadrics in [7] intersecting in curves C ~4 and C ~~ of orders 14 and 50 respectively, the former being the canonical curve of

4 0 4 L . s . G O D D A R D

genus 8. And m is equal to the number of intersections of these two curves. We use the result ($) that in [r], if r - - 1 primals, of order ~t , . . . , p,._~ intersect in two curves of order ~o, ~'o and ranks ~, ~'~ the number m of common points of the two curves is given by

�9 i -Jr- m --- eo Or r + m : - e'o ~, where a - - y ' (pj - - 1). J

The rank e, of a curve Cp ~ of order n and genus p, is given by e - - 2 (p + n - - 1). Thus, for the canonical curve C'~', we have r = 42, and m : 42. (We also find that C ~~ is of genus 80, and rank 258). It follows finally that m == 42, and [8]-~ V~ ~ cutting ~(~) in a V~'.

Suppose now that, for I ~ d ~ 7,

[d] . ~ V ~ ' 8 - d cutting fl(~l in a v nS-e --d--1

where the numbers ms_d and ns_~ are to be determined. Between these numbers we clearly have the relations

m~ = 2 m~_~ - - nj_~ ( j = 2, 3 , . . . , 7)

and also ml - - 100 - - m : 58. But m e , . . . , m7 may be calculated thus: to a

[d] there corresponds a V~'~8-~ and the order ms-~ is equal to the number of points common to V~ and a [14 - - d]. On using the fact that T~ is symmetric we see that ms-~ equals the order of the Vt4-~ which corresponds to [ 1 4 - d]. Thus m2, . . . , m7 have the values 50, 32, 16, 8, 4 and 2 respectively. The above relations then yield for n , . . . , n6 the values 66, 68, 48, 24, 12, and O.

It remains to find n~, which is the number of points common to f~(~) and the conic which is the transform of a given line. Denote the line by l and the conic by y. Under the transformation, to a point w (2) of Q(~) there corresponds a [5], w (4) o f ~'~(4) Thus, the number of points common to ,f and s equals the number of generators w (4) of f2(4) which meet the line 1. This number is 3, since s (4) i s a cubic primal, and hence n7 : 3.

9. - - The Case of a Skew Matrix of Odd Order. When S is of odd order adj S is symmetric. Also adj S is of rank 1, since S is singular, and thus adj S ~- O. Thus it is not possible to set up a Cremona transformation, of the type we have been considering, in this case. it is instructive, under these conditions,

CREMONA TRANSFORMATIONS IN THE GEOMETRY OF MATRICES 4 0 5

F 1 -I to consider the transformation of the p a r t i c u l a r [ - ~ n ( n - - 1 ) - l l g i v e n b y x ~ j +

L - - J

-~- xj, --- 0, in the general transformation between two [n 2 - - l]'s, defined by

Y = adj X, ~ X :-:- adj Y.

Let Z~ and 2:z be two [4 r (r + 1)]'s in which the coordinates are the ele- ments of generic matrices X and Y, each of order 2 r + 1. Denote by v . the subspace [ ( 2 r - - 1) (2 r + 1)] of ~ given by x~j + xj, = 0, and by Y~* the subspace [ r ( 2 r - + - 3)] of "~2 given b y y o - - YJ , - - 0. Let S be the skew ma- trix whose elements are coordinates in Z~* and (3 the symmetric matrix whose elements are coordinates in Z2*. We consider the transformation

(3 - - adj S.

If we denote by A t , . . . , A2,+~ the Pfaffians whose squares A ~ , . . . , A~,.+~ are the principal minors of S of order 2 r we have, by an algebraic proprierty of skew matrices,

(aaj S),j = a, aj,

so that (3 : adj S = c c',

where c is the column vector {A~ , . . , , A2,+tl. Now let u : { u ~ , . . . , u~+~l be prime coordinates in a space P~, and consider the correspondence between primes of P~r and points of Y't*, defined by

S u = O

If u is given, the 2 r - q - 1 primes so defined in Y~t* cut in a [ ( 2 r + 1) (r - - 1) - - 1], while, if S is of rank 2r (and not less), that is, S is a given point not on the locus defined by adj S : O, the equations have the unique solution

o U~ - - A .

where a is a constant. Thus there is a (1,1) correspondence between the points of P~ and the members of an c~'z~ system R of [(2 r -+- 1) (r - - 1) -- 1]'s in ~t*. The system R possesses the property of having a unique member passing through a generic point of Y,~*.

4 0 6 L. s . C O D D A . D

Using this correspondence we have

{ 3 : a , j S = c c" --- "c u u',

where -c is a constant, so the points (3 of Z~* given by {3 = adj S form the variety

y(l) - - (S 12 r -1- 1, 2 r -~- 11,, [r (2 r -+- 3)]),

which [(1), p. 135] is the Veronesean of quadrics in [2 r]. Hence in the transfor- mation {3 - - adj S between ~,~* and E~* to a [(2r + 1) (r - - 1) - - 1] of the system t? in ~,~* there corresponds in ~'2" a point of the Veronesean Y").

T h e t r a n s f o r m a t i o n T*

10. - - T* is a transformation, by primals of order n, between two [n~]'s, Y~* and ~ * in which the coordinates are (X, X) and (t~, Y) respectively. By writing T* in the form

--IXE+Xl

I~E !- Y--~ 2 ~. adj O. E + X),

this seen that the fundamental points satisfy either (i) 7. - - ~ - - 0 or (it)

adj 0. E + X) : 0. Denote by uTthe prime in Zl*given by ). : 0. The points (i) form the chain of key-loci

K (i) C K (2) C . . . C K ('~-i) --" K,

in ~,, and the points (it) the chain

D (~ C D (~) C . . . C D ('~-~) - - H (say),

where D (~) is defined by the vanishing of all minors of ~ E + X which are of order r + 1, that is I ~ ' E + X[r : 0. It was proved in a recent paper (7) that D Irl is the cone joining the point D/~ given by 7. - - - - 1, X - - E t o K<~) and that the tangent space to D I~) at a point P touches D/~l at every point of an [r~], t*(~t. We shall refer to the spaces t*(~/ below, in discussing the transforms of the fundamental points.

C R E M O N A T R A N S F O R M A T I O N S I N T H E GEOMETRY OF M A T R I C E S 407

Consider now the multiplicity of the point D (~ on the primais ~. and y~ where Y - - (Yo). If ()., X) is the generic point of ~2~*, the generic point of the line joining (Z, X) to D (~ has for its coodinates (-- ~ -I- ~ ~. ~ E - I - e X). Sub- stitute these into the equations of p, and the y,j. Then

rt - - I ( - - ~, -I- ~x) E q - ~ E - I - e X I - - I~ (x E - t - X) l --- e'~ I x E q - X[,

IL E -q- Y = 2 ( - - ~ -~- s ~ ) adj { e ( ~ E ~- X ) }

- - 2 ~-~ (-- �9 .-]- e ~.) adj Q. E -q- X).

Thus D (~ is n-pie on tt and (n - - 1)-pie on y~. From this it follows that D (~ is an (n - - 1)-pie point on the general primal of the system defining T* and hence T* is a monoidal transformation. Further because of the symmetry between T* and T *-~, T* is bimonoidal (for these terms see (8), p. 306).

We may now summarize these results in

T h e o r e m 5: T ' i s a bimonoidal transformation between two [n2]'s by a

system o[ primals of order n passing through the "primal, K in rv, which is of dimension n ~ - 2 and order n, and the cone H, of dimension n ~ - - 4 and

order 1 n~ (n ~ _ 1), joining the point D (~ to the double locus t f (~-~) of K. 12

1 1. - - Transformation of the Fundamental Loci of T*. As in w 3 we denote the fundamental loci in Y~t* and ~ * by the subscripts 1 and 2 respectively. These loci transform as follows:

K~ (t)-~ I~, Kt (~).>. De (~-~) (r = 2, 3 , . . . , n - - 1)

Dt (~ ->- w~, D~ (~) -~ K~ ('~-~) (r - - !, 2 , . . . , n - - 2)

with similar results for the transformation from ~ * to Y~I*. More detailed results are contained in

T h e o r e m 6: (i) To a point Xo on K~ (~) there corresponds a [ ( n - r)~], t~ *<~-~), on D~ (~-~) and to each point of the [r ~ - - 1], tt (r), which lies on If~ (r) and passes through Xo there corresponds the same t *r In particular, the points of 1(t (~) are transformed into the generators o f the cone ~ .

408 L. s. ~.ODDAI~D

(ii) To a point (Z, X) on Di I') there corresponds a [ ( n - - r ) 2 - 1], t I~-€ on K~ ~"-'), and to each point of the [r~], t*<~) which lies on D~ ~) and passes through

(~, X)there corresponds the same t <~-~t. To the vertex D~<oJ there corresponds

the prime {vz. Proof: (i) Consider the t~ I~) given by

where P is an arbitrary r i.~( r matrix. On substituting Xo-i- * Q. E + X)into the equations for I~ and Y, and taking the limit as , -~ O, we find that

p . - - [ ~ . E , _ ~ - ~ - X [ , t ~ E + Y = ( 0 0 ) 0, 2 ~. adj (~. E,_~ + X-) '

n

where X is an arbitrary matrix of order n - - r. It follows that j 1~ E -+- Y Is-,. = 0, and therefore (I ~, Y) is a point on D~/"-~/. Moreover, the totality of points (F, Y) consists of a G *("-~/, which is the transform of an [ ( n - r)2], by a transforma- tion of the type we are considering. Also this t~ *c'~-') depends only on the t~ Irt, and not on the particular point chosen in t~ (r) (since the factor I P [ has been removed). Hence part (i) of the theorem follows. (if) Consider the t~* which is the join of D/0/ to the t It/ given by

where Q is an arbitrary r X r matrix. Since D/0) has the coordinates ;~ = - - 1 X - - E, it follows that the generic point of t / is

~ ,=~ ,o X _ _ X o _ _ ( - - X o E,_, 0 ) ' o O - - X o E , "

If the coordinates of the point, ~.o/~- -I- Xo -[- e (Z E-~ -X) be substituted into the equations for I~ and i1, it is found that V, and Y commence with terms involving ="-~ and r . . . . i respectively. On removing the common factor and taking the limit we obtain

CRE1MiONA TRANSFORMATIONS IN THE GEOMETRY OF MATRICES 4 0 ~)

where X is an arbitrary matrix of order n - - r. This is a point on K ~_~ .... ~., and for varying ()., X) we obtain a t.2 I'~-~), which, as before, is independent of the particular point chosen in t, */'). In w 10 it was shown thai D (~ is n-pie on I ~ and (n - - 1)-pie on the primais y~j. Hence D (~ is transformed into p. ~ 0, Y---

--- adj (~ E + X) in the prime ~,e. The proof of part [ii) of the theorem is n o w complete.

12. - - Examples : (i) n : 2. T* is the well known transformation between two [4]'s by quadrics th rough a quadric surface and an isolated point.

(it) n : 3. T* is a transformation between two [0]'s by cubic primais, ~ , passing through the V7 ~, K, and the cone/-/ , which is the join of a point D (~ to the 1,'46, K, (u the latter being double on K,. K, consists of ~4 [3]'s, tt ~3), and Ht of 004 lines

�9 .,, *(i) tt*(t) t, */'l These transform thus: t,/31 ~- ,,, , -~ tem/. The cone It, (or IX~)consists of 00 4 [4]'s, t, *(~/ (or t.~*(2/), and the points of K, !'~ transform into the [4]'s, t~ *(~) o f It,~.

Concerning the transformation of linear subspaces of the ambient space suppose that (s)

[d] "~ V~ d culling /4 in Va a,_ and K in Va a ,

Now & is the number of points common to V~ a and a [9 - - d], and T* is

symmetric. From these facts it is easy to deduce that

(1) l,l z l:,--d (d ~ 1 , . . . , 8)

Also, m a is tile number of points common to V~ a, and a [10 -- d]. Under the

V ta~ and transformation the It0 - - d] -~ - . , - a

"a v " '~ (on K). Va_ , (on H) -).- -9 a Thus

(2) m,, = n,,, ,, (d =-= 3, 4, 5)

The relations (I) and (2) are used below. We now determine the numbers Id, m,,, n,+. To an [8] there corresponds a

cubic primal �9 containing both H and K. To a [7] there corresponds a V7 6 (the order is 6 since the V; :+, /4, is part of tile base) which contains H and cuts K in a Vo"~. To determine n7 take the section by a [3]. We obtain two cubic surfaces intersecting in a cubic and a sextic curve, and the cubic is plane (since

410 ,.. s . o o ~

the base l o c u s / ( is in [8]). Thus the sextic cuts the cubic in six points, and n~ = 6. To a [O] there corresponds a V0~% which is the residual intersection of V76

and a primal r Each of these contains the V,~ ~ (section of K by Vv ~ and hence /6 -a t- 6 --- 18, that is, l~ -- 12 The variety V,~ ~ contains /-/ and cuts K" in a Vj~6. To find n6 take the section by a [4], n. We obtain 3 cubic primals in [4] intersecting in a cubic surface F, a curve C of order 12, and 6 points, these being the sections by r~ of K, I/6 ~ and l-/ respectively. Now no equals the number of points common to C and F; and this number is 12, since F lies in a solid. Thus no ~ 12.

To a [5] there corresponds a Vsq cutting H in a V4"s and K in a V4"5. It follows from (2) that m . , " - - n~,. The number l.~, is obtained thus : V~5 is the residual intersection of V6 ~ and a primal @, and each of these contains the V5 ~ and the V~ ~ which is the section of K by V,~ 1~. Thus t:, -q-6-q- 12 ~ 36,

that is 1~ -.= 18. To find n~ take the section by a [5], w. We obtain in ~,, 4 cubic primals intersecting in a V:~ ~, and curves of orders 18 and O, these being the sections by n of /(, V; '~ and /-/respectively. it follows, as before, that n~, being the number of points common to 1/33 and the curve of order 18, is equal to 18.

The relations (1) and (2) may now be applied to find the orders of the relevant varieties in the transformation of a [d], for d ~ 1,2, 3, 4. Thus to a [4] there corresponds a i /4 ~8 cutting H in a I/3 ~. Also V4 ~ cuts K in 6 solids; for the [4] meets H in 6 points and each point is transformed in a [3], t,~/3) on K. Next, to a [3] there corresponds a V3 v' cutting /4 in a sextic surface and not meeting /4" (outside the double locus). And, to a plane there corresponds a sextic surface meeting H in 3 lines (since the plane meets K in 3 points), but not meeting K. Finally to a line there corresponds a twisted cubic not meeting H or K.

The transformation T~* defined by a symmetric matrix.

13. - - If (~l and (~2 are symmetric matrices, each of order n, the transfor- mation Vl* defined by

= E + (3, i, G,., = O, E - - (30 aaj O, e + 0 , ) ,

1 ]'s, and in is a Cremona transformation between two ~- n (n -]- 1) ~t*' Y'.~*'

which the coordinates are (;~, (30 and (8, G~) respectively. The fundamental elements in Zt*' consist of the chain of key-loci

Y(J) C Y!~)C . . . . C y ( , ~ - l ) = y,

CREMONA TRANSFORMATIONS IN THE GEOMETRY OF MATRICES 41 1

in the prime ~,, given by ). : 0, and

E/~ C E (~) C . . . . C E ( ' -2)

where E ~') is the cone joining y/,) to the point ). = - - 1, ~1 = E. The equa-

tions of E I") are ] ;~ E -~- Gj !~ - - 0, and 1~ is the cone E I"-l). We recall that yr

- - " u" and hence consists of a single system of ~ / . . . . ) (r @ 1) 1 '~,

E(~) consists of ~l"-"~ l l r (r -~ l) 's, say v!~).

The transformation properties of the fundamental loci are very much like

those occurring in w 11. The results are contained in

T heorem 7: (i) 70 a point (0, •,) on Yl Cr' there corresponds a l l (n - - r )

( n - r -~-1)] , v~ (~-~), on E2 (~-~1, and to each point of the unique [-~- r (r- L- 1) i

1], u Cr), which lies on y l~) and passes through (0, ~ ) there corresponds the J

s a m e V~ ('-' 'j.

(it) To a point (X, Gt) on E, r there corresponds a ~ I 1 (n - - r ) (n - - r ~- 1 ) - - 1

�9 1 11 L ue ('-~), on Y2 ( .... ) and to each point of the unique ~ r (r 4- , v~% which

lies on E~ ('i and passes through 0', G~) there corresponds the same u~ ! ..... ~. To

the vertex E (~ there corresponds the prime ~v, z. This theorem is proved by methods similar to those used in the proof

of Theorem 6.

Examples: ( i ) n - - 2 . Tt* is the well-known transformation between two [3]'s by quadrics th rough a conic and a point.

(it) n = 3. Ti* is a transformation between two [5['s by cubic primals, r con-

taining the I/4 :~, ~, and a point cone, E (~/ - - I/3 4, joining a point not in ~7 to the Veronese surface y~l! which is the double locus on Y. Y consists of oo~ planes, u% )"~/ of toe points, u% and E (~) of oo~ lines v (1). The cone ~ ( = E (~))

412 J.. s . ~;.aaARD

consists of oo 2 solids, v 12), and the fundamental spaces transform thus :

11t (2) ..~ 1)2(t) , l / l ( t ) . ~ U2(t)~ //t (1) .~ 1)2 (2).

By m e t h o d s similar to t hose used in the p reced ing examples we derive

these results:

(i) to a [4] there cor responds a V46 containing the cone E ~tl and cut t ing Y in a I/36

(it) to a [3] there cor responds a V3 s cutting E It/ in a sextic surface and cut-

ting Y in 4 planes. Since the [3] c u t s the base in a twis ted cubic and 4 points

it follows that V3 s is the Veronesean o f cubic sur faces through a twis ted

cubic and 4 points .

(iii) to a plane there cor responds a sextic del-Pezzo surface (the Veronesean of

plane cubics t h r ough 3 points) cutt ing E (t) in 3 lines.

(iv) to a line there cor responds a twisted cubic.

Aberdeen, Juin 1953

N O T E S

(~) The subscript I refers to loci in ~'L, and the subscript 2 is used for the similar loci in [~.

(-~, There is no loss of generality in taking the ~i (~) to be given by the matrix (0 p 0~ 0]" It is

shown in my dissertation (~, p. I1) that, by a suitable choice of the coordinate system, it is possible to parametrise the spaces L ~>, A'/('>, [L c~), ..ff(~)] and t C'~ by the matrices

(0" a.d (0 respectively, where P, Q, and R are matrices, of orders r X r, r X (n -- r) and (n - - r} X r, whose elements are all independent.

(3) From this point onwards we drop the suffixes I or 2 on /-/ aud K since the correct suffix is clear from the context.

CIqE.&IONA ' I ' I I A N S F O R ' M A T ! G N S I N T i l l . G E O M I : T I I Y (IF .Mh 'FRICES 413

R E F E R E N C E S

(1) T. 13. RooM, The Geometry of Determinantal Loci (Cambridge, 1938). (2) O. Rozt~T, Bull. Sac. Roy. Sci. Liege, 8 (1939), 86. (31 O. Rozm'r, Bull. Sac. Roy. Sei Liege, 8 (1939), 367. (4) L. S. OOODaRD, The Oeomary of ,Matrices (Cambridge Dissertation, 1946). 15) H. F. BAKa~, Principles of Geometry. VI ~Cambridgr 1933), 251. (5) O. Rozs'r, Bull. Soe. Roy. Sci. Liege, 8 (1939), 12. ~7) L. S. OODD^aD, Proe. Cambridge Philos. Soc. 47 (1951), 286. (8) H. HUDSON, Cremona Transformations (Cambridge, 1927~.

27 - R e n d . C i r c . 3 1 a t e m . P a l e r m o , - - s e r i c I I - t o m o 11 - a n u o I953