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HERMITIAN VARIETIES IN A FINITE PROJECTIVE SPACE FG(N, q2)
R. C. Bose and I. M. Chakra.varti
University of North Carolina.
Institute of Statistics Mimeo Series No. 428
May
1965
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This research was supported by the Na.tional Science
Foundation Grant No. GP-3792 and the Air Force Office
of Scientific Research Grant No. AF-AFOSR-76o-65.
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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Hermitian Varieties in a Finite Projective Space PG(N,q2)
R. C. Bose and I. M. Chakravarti
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University of North Carolina
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The elements
one another.
x
and x
q
of G.F(q2) are defined to be conjugate to
= «hij » is called Hermitian if hij
If !.T:: (xo,~, ... , ~T)' then !.(q) is
The square Il8trix H
is conjugate to h
ji for all i, j.
defined to be the colwnn vector whose elements are
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present paper studies the geometrical properties of Hermitian varieties.
The theory of pole andpolars has been developed, and the sections of these
The number of fiat spaces of
a given dimensioo. contained in a Hermitian variety has been obtained.
.
q+l
q+l
q+l
q+l
FJ.nally the geometry of the surface xo + ~ + ~
+ ~
= 0 , has been
studied in some detail leading to a geometric interpretation of same designs.
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q
q
:Ki'
••• , XJ.Ii.
Hermitian variety V _ in the finite projective space PG(N, q2) has the
Nl
equation !.T H x(q) = 0, where H is a Hermitian matrix of order N+l. The
varieties by hyperplanes have been studied.
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xo'
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HERMITIAN VARIETIES IN A FINITE PROJECTIVE SPACE PG{N, cf)
R. C. Bose and I.
~
Chakravarti
University of North Carolina
-----------------------------------------------------------------------------1.
Introduction.
The Geometry of quadric varieties (hypersurfaces) in finite projective
spaces of N dimensions has been studied by Primrose (12) and Ray-Chaudhuri (13).
In this paper we study the geometry of another class of varieties, which 1m
call Hermitian varieties which have many properties analogous to quadrics.
Hernutian varieties are defined only for finite projective spaces for which
the ground{Galois field) GF{q2), 1ms order q2, where q is the power of a prime. If h
is any element of GF{cf), then
2
her
= h,
h is conjugate to
ii.
ii = h q
is defined to be conjugate to h •
A square matrix H
= ({h J.J
.. »,
wi tl1 elements from GF (q2) is called Hermitian if h ij
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set of all points in PG{N, cf) 'Whose row vectors !:.T
the equation
~?H ~(q) = 0
= h j i'
i,j
Since
= O,l, ••• ,N
for all i, j.
= (xo'xl , ... , XN)
The
satisfy
are said to form a Hermitian variety V _ , if
Nl
H is Hermitian and )q) is the column vector whose transpose is
(x5'
xi, ..., 4-).
The properties of the curve
1
1
xci"+l+ 4+ + 4+
=0
in
PG(2,q2), Which is a Hermitian variety, were studied in some detail by one of
the authors in
sions.
(3).
The present paper generalizes these results to N dimen-
The theory of tangent and polar hyperplanes of Hermitian varieties
has been developed, and the sections of these varieties by hyperplanes have
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been studied.
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This research was supported by the National Science Foundation Grant
No. GP-3792 and the Air Force Office of Scientific Research Grant No.
AF-AFOSR-76o-65.
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The number of points on a Hermtian variety has been obtained.
It has been shown that if N
= 2t+l
or 2t+2 , a non-degenerate Hermitian
variety V _ contains flat spaces of
N l
t
dimensions and no higher.
The
number of such subspaces contained in V _ has been derived. Finally the
Nl
q+l
q+l
q+l
q+l
.
geometry of the surface Xo + ~
+ X2
+ X-j
= 0, has been studied l.n
some detail, leading to a geometric interpretation of some designs.
example if q
= 2,
For
the surface contains 45 points and is ruled by 27 lines
Corresponding to any point P on the
three of which pass through each point.
surface we get a set of 12 points which are joined to P by a line on the
The 45 sets so obtained form the blocks of a balanced incomplete
surface.
block design with parameters
v
= b = 45,
= k = 12,
r
A = 3.
There are many
other interesting designs and configurations connected with Hermitian
varieties.
2.
These will be discussed in a separate connnunication.
Correspondence between the. elements of GF(q) and GF(cf).
Let q
= pm
, where
p
is a prime number and
Let GF(q) be a Galois field with
GF(q).
If
e is
q
elements, and GF(q2) an
extension of
are
,.2
••• , ecf -l = 1
e,t:1,
Any non-zero element
•
x of GF(q2) satisfies the f'undamental equation
2 1
xq -
= 1
Writing
(2.3)
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it follows that
(2.4)
-.
m is a positive integer.
a primitive element of GF(q2), then the elements of GF(q2)
O,
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O,
...'t', CV-,
~
•••
2
,
...'t'q-l
=1
,
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are all different and are elerents of GF{q).
of G]'{q), and all the elements of G]'{q) are given by
Corresponding to a given element
element
q
-I-
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y
y
= x q+l
is a primitive element
(2.4).
of GF(g,2), there is a unique
•
belonging to G]'(q)J there are precisebY
1 distinct elements x of GF(q2) which satisfy (2.5).
Thus if
y = 4>i
(2.6)
= ;(q+1)
,
1
~
i
~
q-1
then
x
(2.7)
If however
y
= ei +j (q-1)
,
j
= 1,2,
••• , g,+1 •
is zero, tl1en the corresponding element
x
of G]'(q2) is
zero.
3.
Conjugate elements of GF({).
The primitive element
e of
GF{q2) satisfies a quadratic equation
x2-sx+t=o,
\'There
s
and
t
belong to GF{q), and the left hand side of
(3.1)
is
irreducible over GF (q).
Using the relation
rP- - s e + t = 0, every element x of GF (q2) can
be expressed as
x
Where
a
and
b
belong to
=a
+
G]'(q).
be ,
We then define
,.
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x
cl>
belonging to G]'(q), given by
But for a given non-zero
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y
(2.5)
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Hence
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as the conjugate of
x.
~5.L/-)
x
the conjugate of
-,
Since
~.
x is
If
q2
x
=x ,
is given by
(3.3),
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then
x = (a+bo)q = a+bOq = a+be •
(3.5)
q
x -.x is an automorphism of GF(q2), the second root of (3.1)
Since
is oq or
O.
Hence
(3.6)
O+o=s,
x +
x = 2a
+ bs ,
(3.8)
Hence the sum as well as the product of two conjugate elements of
GF(q2) belongs to GF(91.
It should be noted that the necessary and sufficient condition for
any element of GF(q2) to be self conjugate is that it belOI'lBS to GF(q).
The elements s
From (3.6), t
non-zero.
Again if
o =0
s
= 0,
or 0'1-1
/'1- 2 = 1
and
t
= 0'1+1
of GF(q) appearing in the equation (3.1) are
1=0 since 0 is a primitive element of GF(Cl2 ).
it would follow from (3.6) that 0 + 0'1
Obviously 0 =I O.
= -1.
= 0,
i.e. either
of GF('12) •
non-zero element
Let h
r.
= a+bO
If
h
is a non-zero eleroont of GF('12), we can find a
of GF( '12 ) such that
and
r. = u+vO
h~ +
, where
hr. 1=
0 •
a,b,u,v belong to GF('1).
using (3.6)
h~ +
hr. =
(2a + bs)u + (2bt + as)v.
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Also oq-l 1= -1 , otherwise
, which is contradicted by the fact that 0 is a primitive element
Lennna. (3.1).
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Then
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If
2a + bs
Case II. If
2a + bs
Case I.
contradicting h
F O.
F 0,
= 0,
we can choose u
then a
Now (2bt + as)
is the condition for the roots of
OQ. ,
= a(s2
(3.1)
Which is obviously false since
F '0,
e
= 1,
=0
since a
- 4t)/s
v = 0, i.e. A = l-
F0
= 0,
, since s2 - 4t
to coincide i.e. for
e to
=0
be equal to
is a primitive elenent of GF(q2).
Hence in this case we can choose u = 0, v = 1, i. e. A =
4.
"TOuld make b
e•
Herm1tian matrices and Hermitian forms.
A square matrix
(4.1)
= 0,1,
i,j
••• , N
with elements from GF(q2), will be defined to be Hermitian if
(4.2)
for all i, j.
Hence the diagonal elements of a Hermitian matrix belong to
GF ( q) , and symmetrically situated off diagonal elerents are conjugate to
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each other.
Given a matrix A
= «aij » with
elements f ... om GF(q2) we define the
conjugate of A by
A (q)
(4.3)
-
«aij q»
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Clearly, the conjugate of A(q) is A itself.
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a square matrix.
•-
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is symmetric.
-( (n
-
ij
»•
Thus the relation of
conju~c~
So far as tllis definition is concerned A mayor may not be
In particular A may be a row vector or a column vector.
Clearly, the necessary and sufficient condition for A to be self conjugate
is that all its elements belong to GF{q).
The transpose of A will be denoted by AT.
Clearly the transpose of
the conjugate is the conjugate ot: the transpose, i. e •
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AT(q)
(4.4)
Lemma. (4.1).
= A(q)T
A square matrix G
=
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«g..J.J »
with elements from GF(q2)
is Hermitian if and only if'
The proof' is obvious.
Lemma. (4.2).
n.."C11
Suppose
A and B are two matrices of' order
with elements from GF(cf), and C
Proof:
.
..
Now C
_
cik
E
«c ik
»,
c ik
= E aijb jk
q
c ik
=
Hence by definition
Lemma
=
(4.3).
j
= AB,
mxn
and
then
Where
•
(n
~=laijbjk
)q
n q q
n_ _
= j:1aijbjk = j:1aijbjk
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C(q) = A(q) B(q) •
If'
H is a Hermitian matrix of' order N+1, and A is
any matrix of order (N+1)x m "lith elements from GF(q2), then
G = AT H A(q)
is a Hermitian matrix of order
~
,
•
From Lemmas (4.1) and (4.2), and the equation (1~.4),
GT
= A(q)T
HTA = AT(q) H(q) A
= G(q)
•
The required result follows from Lemma (4.1).
Corollary.
matrix of order
Proof.
If ~ is a (N+l)X 1 column vector, and H is a Hermitian
N+l
then
~TH 2f(q)
~TH ~(q)
is an element of GF(q) •
is a 1 x 1 Hermitian matrix.
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Hence it is a self'
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conjugate element of GF(q2) •
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The elements of all the vectors and matrices w'hicl1 He shall consider
belong to GF{c'f).
vectors ,·re shall mean dependence and independence over crF(q2).
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The rank of
a vector space or the rank of a matrix shall mean rank over GF(q2).
Two Hermitian matrices
H and
G of the same order N+1 with elements
fioom GF(q2), will be called equivalent if we can find a non-singular square
matrix A, \'lith elements frOI';l crF(q2) such that
AT H A (q)
If
H and
=G
•
G are equivalent we may write H .... G.
It is readily seen
that this relation satisfies the three axioms of equivalence L e. (i) H .... H,
(ii)
if H .... G, then G .... H, (iii) if H .... G and G.... K, then H .... K •
The above follows by noting that (i) IT = I(q) = I "mere I is the
(AT )-l = (A- 1 )T , (A(q»-l = (A-l)(q)
unit matrix of order N + 1 (ii)
(iii)
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When ,.,e speak of the dependence or independence of a set of
BTAT
= (AB)T
, A{q)n{q)
Theorem (4.1).
= (AB)(q)
from Lemma
(4.2).
A Hermitian matrix H of order N+l and rank r > 0,
,nth elements from GF(q2), is equivalent to a diagonal matrix of order N + 1,
the first
(a)
r
diagonal elements of which are unity and
We can pernmte the columns of
tl~
rest zero.
H in any desired manner and
permute its rows in the corresponding manner by post multiplying
suitable permutation matrix P
= p(q),
and prenmltiplying
H with
H with a
pT.
Hence by such operations "le can rearrange the rows and columns of H
all null rows and columns are at the end.
to
so that
The transformed matrix is equivalent
H.
(b)
We shall denote by E (A), a matrix of order N+l, for which each
uv
diagonal element is unity, the element in the u-th rOvl and v-th column is A,
and all other elements are zero.
Such a matrix will be called an e1ementa.ry
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matrix of order
N+1.
-,
Clearly
T (A)
Euv
= Evu (A).
The effect of pre:multiplying
H
with
ET (A) is to replace the v-th
uv
ro't'7 of H, by the sum of the v-th row and the u-th row multiplied by A
The effect of post multiplying the matrix so obtained with
E(q)(A) is to
uv
replace its v-th column by the sum of the v-th column and the u-th column
multiplied with ~.
Thus if
H
is given by (4.1),
ET (A) H E(q)(A)
uv
uv
= G = «giJ.»
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1-There
fl.
l=hv-l,v-l +Ahu-l,v-l +~hv-l,u-l ,
-v-l,v-
(i
If the v-th row and column of
element
h
h
v- 1 ,v- 1 = 0, then
Fv-I,
j
Fv-l)
•
1
1 and h 1
1 belonging to the (u-l)-th row and column respectively.
u- ,vv- ,u-
By Lemma
Ah
(3.1) there existfl
u-1,v-1
+
~
h
v-1,u-l
(c)
Then the matrix ET (A) H E(q)(A) is equivauv
uv
F v.
lent to H. and the element
~
element A of GF(q2) such that
'til'"
g
-v-
1
,v-1
in the v-th r01-T and column is non-zero.
By using (a) and (b) suppose
H has already been transformed to
an equivalent form such that the first row and column are non-null and
h
OO
FO.
We now reduce the non-diagonal elements of the first row and
column to zero, in
N steps, the (v-1)-th step consisting of premultiplying
the matrix obtained in the previous step by Eiv(-hv_1,dhoo) and post
multiplying it by
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H are non-null but the diagonal
can find non-zero conjugate elements
vie
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Ei~)(-hv_l,c1hoo)'
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v = 2,3, ... , N+l.
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If any null rows and columns appear they are transferred to the end by
using (a).
If now the diagonal element in the second row is zero, (and the
second rm'T is non-null) ,ore can make it non-zero by using (b).
Then as in
( c), we can reduce the non-diagonal elements of the first row and column to
zero.
Proceeding in this manner we reduce
H to an equivalent diagonal
matrix D, in which the first r 0 diagonal elements are non-null, and the
remaining diagonal elements are null.
rank-preserving r
(d)
= r.
o
Since
D is Hermitian the diagonal elements belong to GF(q).
Let the i-th diagonal element be d
section 2, we can find an eJ.ement
(i
= O,J.,
••• , r-l).
diagonal eJ.ement is
finally reduce
Since all our transformations have been
•
i
a
From the correspondence described in
of
i
GF ( q2) such that d
:: a q+l
i
i
= a i -a i
We denote by A(a ) the diagonal matrix, whose i-th
i
a. and. the other diMonal elements are zero.
We can
J.
D to the form desired in the theorem in
r
steps, the
(i+l)-th step consisting of premultiplying the matrix obtained in the
previous
(i
= 0,J.,
Let
step by
DT(a.)-l
••• , r-J.).
T
.?f
and post multiplying it by D(q) (a.)-l,
1
This completes the proof of the theorem.
be the row' vector (xo,x:t, ••• , ~), and! the correspondinG
column vector, Where
~T
1
x o ' Xl' ••• ,
XN
are indefinites.
H 2S(q) is called a Harmitian form, if
ca.lled the matrix of the form.
to be the order and rank of H.
H is a Hennitian matrix.
Note that
xTH x(qJ
is a homgeneous
xO,x:t, ••• ,
"N •
is transformed into '1.T AT H A(q) '1.(q)
The Hermitian form !T H 2S(q)
~
H is
The order and rank of the form are defined
polynomial of the (q+l)-th degree in the indefinites
by the linear transformation
Then the form
= A '1..
Two Hermitian forms are defined to
be equivalent if one can be transformed to the other by a non-singular linear
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transformation.
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Clearly the necessary and sufficient condition for two
Hermitian forms to be equivalent is that their matrices are equivalent.
Corollary The Herr.rl.tian form xT H x(q) of order N+l and rank r
be reduced to the cano.nical form y1Y1 +
non-singular linear transformation
~
Y
Y2 2
+ ••• + YrYr
can
by a suitable
=A~ •
5. Hermitian varieties in PG{N,q2)
We denote by PG{N,s) the finite projective space of N dimensions
over the Galois field GF(s) where
s
is a prime pOi-rer.
The points of the
space can be made to correspond to ordered (N+l)-plets
(xo'ZJ.' ... , ~) ,
vnlere the
x.1 's belong to arCs), and are not all zero.
The ordered n-plets
*
*
(xO,Xl , ••• , ~) and (x*
O' "l ' .•• , ~ ) correspond to the same point if and
only if there exists a non-zero element p of GF{s) such that
p
x.*
1
= x.J.
i
= 0,1,
••• , N •
If P is the point corresponding to (5.1), then the row vector
x
T
= (Xo'~1.'
••• , ~) is called the row vector of P, and its transpose x
is called the colu"'n vector of P.
The elements
xO,x1 ' ••• ,
~
are called
the coordinates of P.
If C is a matrix vdth N+1 rows and of rank N-m
w~th
elements
from GF(s), then the set of points whose row vectors satisfy
T
x C
=0 ,
is called an m-flat or a linear subspace of m dimensions, and (5.2) is
called the equation of the m-flat.
dimensions.
Points are linear subspaces of zero
Linear subspaces of 1,2 and N-l dimensions are respectively
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called lines, planes, and hyperplanes.
A set of points will be said to be dependent or independent according
as the corresponding row (colunm) vectors are dependent or independent.
m + 1 independent points determine a unique m-flat containing
Let
the~
E be the point for which the (i+l)-th coordinate is unity, and
i
other coordinates are zero (i
= 0,1,
Whose coordinates are unity.
Then EO,E , ••• EN are called the fundamental
l
points and E
E ,E , ••• ,
O 1
~
••• , N).
is called the unit ;point.
, E are independent.
Also let
Clearly any
E be the point all of
N of the
N+1 points
Together they are said to constitute the
reference system.
Let
ares).
A
be an (N+l) x (N+l) non-singular matrix 'tnth elements from
Then the homogeneous linear transformation
~ =
(5.3)
A!
defines a transformation of coordinates.
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of P, the transformed column vector is
fundamental points
coordinates are
If
~.
x is the original column vector
This transformation defines new
FO,F, ... , FN and a new unit point F.
Their transformed
(1,0, ••• 0), (0,1, ... , 0), ... (0,0, ... , 1) and (1,1, ... ,1)
and the original coordinates can be calculated from (5.3).
An important
theorem states that given any N+2 points PO,P , ••• , P and P, no
l
N
N+lof
which are dependent, there exists a unique linear transformation, which
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would mall:e PO,P , ... , P the fundamental points and P the unit point.
N
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fundamental points and the unit point.
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Any
in PG(N, s) any
Thus
N+2 points no N+l of which are dependent may be chosen as the
This choice uniquely determines the
coordinates of all other points (up to a non-zero multiple of ar(s».
Pro-
jective geometry studies tl10se properties which are invariant under linear
homgeneous transfonnations and are thus independent of the choice of a
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reference system.
found in
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An excellent account of finite projective spaces will be
(1,10,15).
In particular let us choose s
= q2,
where
q
consider the finite projective space PG(N,q2).
If
is a prime power and
H is a Hermitian matrix
of order N+l and rank r with elements from GF(q2) then the set of points whose
coo'rdinates satisfy the (q+l)-th degree equation
(5.4)
=° ,
xTH x(q)
are said to be the points of a Hermitian variety V _ of N-l dimensions and
Nl
rank r.
The equation
(5.4)
,is said to be the equation of V _ •
N l
the 'inear transformation
(5.3)
(5.J)
~T
If we apply
the new equation of V _ becomes
Nl
AT
HA(q)~ = ° .
Now AT H A(q) is a Hermitian matrix of rank r equivalent to H.
Hence
the ranJr "'If a Hermitian variety is invariant under a non-singular linear transformat:... m.
It follows from Theorem
(4.1)
and its corollary that by a suitable
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choice of the frame of reference, the equation of a Hermitian variety of N-l
dimensions and rank r can be reduced to the canonica' form
-
-
-
xOxO + xlxl + ••• + xr_lxr _l
= °·
A Hermitian variety V _ of N-l dimensions is said to be non-degenerate
Nl
if its rank is N+l.
Now PG(N,~) contains linear subspaces of dimensions r<1f.
Let Z be such a subspace.
r
Then each point of Z
r
set of r+1 coordinates (Yo' yl' ... , yr).
can be characterized by a
For example if we choose the frame of
reference so that the equations of Zr are Yr+l = Yr+2 = ... = YN = 0; then if
the point P, when regarded as a point of PG(N,q2), has the row vector
T
if.
=
(yo'Yl' ••• , Yr' 0,0, .... 0); then regarded as a point of .Er it has row
vector
~*'l! =
(Yo'Yl' ... Yr ).
Then if H* is a Hermitian matrix of order r+l,
the points of Zr which satisfy the equation 1.*'l!H*Y*( q)
= ° 'dll
be said to form
*
the Hermitian variety V 1 of dimensions r-l and rank equal to the ra.nk of H •
r-
We shall in what follows alvlaYs denote
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a Hermitian variety with the letter
sUbscript of
V denotes the number of dimensions of V.
Let us consider tile special case
jective line PG(1,q2).
space.
Let
xoxo + ":J.xl
satisfying (5.7), X
o
F O.
.e
I
=0
q+l
Xo
+
or
q+l
~
=0
•
Hence for points
Now (5.7) gives (":J./Xo)q+l
=-
1.
The correspond-
ence described in section 2, shows that there are precisely q + 1 values of
Hence a non-degenerate Hermitian variet;r V 2E.
o
a projective line (over a field of order q2) contains exactly q+l distinct
Xl/XO
vthich satisfy (5.7).
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I
space is now the pro-
Our
The point (0,1) obvi.::>Usl:y does not lie on V •
l
point satisfying this equation
I
= 1.
Then the equation of V0 can be taken as
I_
I
N
V be a non-degenerate Hermitian variety in this
o
:2Oints.
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V and choose our notation so that the
Again suppose the rank of V0 is one.Then by a suitable choice of the
frame of reference its equation can be reduced to
is (0,1).
l
xci+ = O.
Hence in this case
The only
Yo consists of
a single point.
The properties of the curve
X
q+l
o
q+l
+xl
q+l
+~
were studied in some detail in (:3).
=0 ,
In particular it
v18,S
shown that a
non-degenerate Hermitian variet;r V in PG(2, q2) has exactly
l
6.
q3+l
points.
Conjugate points, polar spaces and tangent spaces.
Consider a Hermitian variety V _ with equation (5.4). A point C ,'lith
N l
T
row vector.£ = (cO' c ' ••• , ~) will be called a singular :2O int of VN_l if
l
£T H
=0
or equivalently H £(q) = 0 •
Of course a singular point must lie on V _ •
Nl
not singular is called a regular point of V _ •
N l
13
A point of V _ Which is
Nl
A point C will be called a
non-singular point if it is not a singular point of V _ • Thus a nonNl
singular point may be a regular point of V _ or a point not lying on V _ •
N l
Nl
A non-degenerate Hermitian variety cannot possess a singular point,
since in this case there cannot exist a non-null
H is non-singular.
Then (6.1) has
If V _ is degenerate let
Nl
N+l-r
r < N+l be the rank of H.
independent solutions
T
(6.1)
S? satisf"<Jing cTH = 0 as
£1 '
T
Ee ' ••. ,
T
~+l-r •
Thus the singular points of V _ are the points of the (N-r)-flat
Nl
determined by the points 'tn.th row vectors (6.1).
This 'tdll be said to
constitute the singular space of V _ •
Nl
.All. points whose row vectors satisfy the equation
xT H c(q) ... 0 ,
'X
constitute the polar space of the point C with row vector .£..
When C is
a singuJ..a.r point of V _ the polar space of C is identical with the whole
Nl
space PG(N, q2). When ho,\,rever C is non-singular the rank of H c (q) is one
and (6.2) is thf' ~quation of a hyperplane, which may be called the polar
hyperplane
of~.
If
T
~
is the row vector of a point D then the necessary
and suf'ficient condition for the polar space of C to pass through D is
T H c(q)
-d
0
-='
which is equivalent to cT H d(q)
-
-
= o.
This shows that
if the "1Olar space of C passes through D, then the polar space of D passes
through C.
Two such points Whose polar spaces mutually pass through each
other are said to be conjugate to each other with respect to V _ • In
Nl
case V _ is degenerate the polar space of C passes through every singular
Nl
point of V _ and thus conta.ins the singular space of V _ • Hence any tvJO
Ul
Nl
points ,at least one of vlhich is singular, are always conjugate to one
another.
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The condition for the point C to be self conjugate i.e. to lie on its
own polar space is that £.T H .£(q) =
o.
with respect to V _ ' if and only if C lies on V _ •
Nl
Nl
The polar hyperplane of a regular point C of V _ is defined to be
Nl
the tangent hyperplane to V _ at C.
Nl
The tangent hyperplane is defined only
for regular points of V _ and when V _ is degenerate contains the singular
N l
N l
space of V _ •
Nl
When V _ is non-degenerate, there is no singular point.
N l
To every
point there corresponds a unique polar hyperplane, and at every point of
V _ there is a unique tangent hyperplane.
Nl
7.
Sections of Hermitian varieties with flat spaces.
Let V _ be a Hermitian variety of rank r in PG(N,q2) with equation
Nl
(5.4).
The set of points connnon to V _l and the m-flat Em with equation
N
(5.2), is defined to be the section of V 1 by E •
Nm
is the whole space and the section is V _ itself.
Nl
be defined by
m+l
independent points
singular linear transformation if.. = A
If m = N, then
Let
FO,F , ... , Fm •
l
~
m < N.
Let
1:
m
Em
vIe can find a non-
such that Fo,F l' ••• , Fm become the
fundamental points of the reference system.
Then the points of
Em will
satisfy the equations
Ym+1
=Y
2
m~
= •••
= YN - 1= 0,
While the equation of V will become
N
(7.2)
,~ere
I
G ~(q)
=0
•
G is a Hermitian matrix equivalent to
,I
Hence a point is conjugate to itself
15
H.
Writing (7.2) in fUll
N
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N
~
~
j=O i=O
~m
Hence the points common to
~
~
j=O i=O
Let
-,
•
and V _ satisfy (7.1) and
Nl
g.
y(q)
j
y
~j
i
G* be the matrix obtained from
m+l rows and columns of G.
=0
gij Yi y(q)
j
Evidently
=0
•
G by retaining only the first
G* is Hermitian and the points on the
section of VN_1by ~m satisfy (7.1) and
Regarding
~
m
asa projective space of
m
dimensions, it is clear that the section of a Hermitian variety V _ in
Nl
PG(N,cf) by a flat space
contained in
~m
•
~
m
of
m dimensions, is a Hermitian variety V 1
m-
Clearly the rank of V 1 cannot exceed m+l.
However
m-
*
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this rank could be less and in particular it may happen that G is null so
that every point of
the secti""n.
~
m
belongs to the section, which therefore coincides with
In this case the flat space
~m
is contained in V _ •
N l
therefore adopt the convention that a flat-space
~m
We will
of dimensions m, can be
regarded as a Hermitian variety V 1 of dimensions m-l and rank zero.
m-
As a particular case let
m
= 1.
Then
~
m
is a line.
Since the inter-
section of a line with V _ must be a Hermitian variety Vo of rank 2,1 or 0
Nl
we see that a line intersects
(iii) lies completely in V _ •
Nl
Theorem (7.1).
~T
H'1S.(q) = 0
V _ in (i) q+l points (ii) a single point or
Nl
We shall now prove the following theorem:
If the Hermitian variety V _ with equation
N1
is degenerate with rank
r < N+l and
dimensions r-l disjoint "lith the singular space IN-r
16
~r-l
is a flat space of
of V _ , then V _l
N l
N
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and Z 1 intersect in a non-degenerate Hermitian variety V 2 contained
r~
rin Zr-1.
Let FO,F1, ••• , Fr _1 be any r
independent points in Zr_1.
let Fr,Fr +l , ••• , FN be any N-r+1 independent points in
a non-singular linear transformation
~-r.
Also
Now make
=A ~
such that FO,Fl , ••• , Fr _ '
1
become the fundamental points of the reference syste~ The
~
F , ••• , F
r
N
equation of VN_1 now becomes "l.T G l.(q) = 0 , where G = «gij»'
(i,j = 0,1, ... , N), is equiValent to H and is therefore of rank r.
y-coordinates the row vector of Fi is ~
Using
for which the (i+l)-th coordinate
is equal to unity and all other coordinates are equal to zero.
for F. and F. to be conjugate to each other is €: G €(q)
=0
The condition
0
gij = •
Since Fr ,Fr +1, ... , FN are singular points of V _ , F i and F j are conjuN1
gate if (i) 0 ~ i ~ r-1, r ~ j~ N (ii) r ~ i ~ N, r ~ j ~ N. This shows
~
-~
J
~j
or
that we may write
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.-
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G =
[~*
"There G* is a Hermitian matrix of order r.
Since the rank of G is
the rank of G* must also be equal to r.
the points of
J!.*T G* X*( q) =
l*T
where
= (YO'Yl'
••• Yr-l).
0;
Now
Yr
Hence
V
r-
2 satisfy
= Yr+1 = ••• = yN = 0 ,
V 2 is a
Hp~tian
r-
variety of rank
r contained in Zr- l ' and is therefore non-degenerate.
Corollary If VN l' L.._ , Z 1 and V 2 have the same meanings
~-r
rras in the theorem, C* is a point on Vr- 2 and Zr-2 is the tangent space
to
V -2
r
~-1
at
c*, then the tangent space to
of N-1 dimensions containing
_ at C* , is the flat space
Nl
V
Zr_2 and
17
r,
~-r
•
Theorem (7.2).
V _ is a degenerate Hermitian variety of rank
Nl
If
r < N+l, in PG(N, q2), and if
C is any point belonging to the singular space
of V _ and D is an arbitrary point of V _ ' then any point on the line CD
NI
NI
belongs to V _ •
NI
Let the equation of V _ be
Nl
!"T H !,,(q)
rOvl vectors of C and D respec tively.
the form
, and lei; £.T and
the
Hence
B is any point on the line CD, then its row vector b
T
T
.tl £. +.t2 ~ or
~ T be
Now C and D are self conjugate, and
also C and D are conjugate to each other.
If
=0
(.tl .£ + .t2~)
T
•
T
must be of
But
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If V _ is as in the theorem and V -2 is the section of
NI
r
V _ by an (r-l)-flat I: _ disjoint with the singular space 1r-r of V _
r l
NI
Nl
then every point of V _ lies on some line joining a point of
N I
~-r
with a
point of Vr-2.
From the theorem if C is a point of E.._.
IN-r
and n is a point of V 2 '
r-
then any point on the line joining CD belongs to V _ "
Nl
Conversely, let Do be any point on V _ "
Nl
We have to show that it
lies on same line joining a point of E.._
with a point of V 2.
IN-r
robviously true if D belongs to E.._
o
IN-r
or I: I"
r-
containing Do and I:r- I"
This is
We may therefore suppose
that Do does not lie on either of these flat spaces"
Let
I: be the r-flat
r
T!len
~in a point C0 and C0 D0 inter. I:r interserts
. J.'if-r
.
sections ~-l in a point po"
fore on Vr-2"
-.
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which proves the theorem.
Corollary.
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From the theorem Po must be on VN_l and there-
This proves the corollary"
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We shall next study the nature of the section of a Hermitian variety
with a tangent space.
We shall first prove the following:
Theorem (7.3).
The tangent spaces at two distinct regular points A
and B of a Hermitian variety V - l are identical if and onl;r if the line
N
joining A and B meets the singular space of V _lin a point. In particular
N
if VN_lis non-degener&te then the tangent spaces at A and B must be distinct.
~t
A and B be
the equation of VN_lbe !,TH
!.T and E.,T.
tions xT H a (q)
=0
- -
~(q) = 0
Then the tangent spaces at A and B have the equa-
and xT H b (q)
--
= O.
Hence the tvTO tangent spaces are
identical if and only if there exists a non-zero element l of GF(q2) such
that
or
First suppose V is non-degenerate.
N
non-singular.
In this case H is
Hence the homogeneous linear equations
= O.
satisfied by £.T
Hence
~? = l(q)E.,T.
£.T H
=0
can only be
The vectors of A and B differ
only by a non-zero multiple of an element of GF(q2).
Hence the points A and
B must be identical.
Now suppose that VN_lis degenerate and of rank r < N + 1.
The
singular space IW-r of VN_lconsists of all points with row vector £.T satisfying
sf H = O.
Hence (7.2) implies that !.T - iq)E.,T
row vector of some point C belonging to 1:_
~-r
E.._
~-r
•
= £T
, where
£.T is the
Hence tm line AB meets
at C.
Coro1l.a.r;l
Let
~-r+l
be the flat space of N-r+l dimensions contain-
ing a regular point A and tbp. singular space 1:_
~-r
variety VN_lof rank r < N + 1.
of a degenerate Hermitian
Then any point B on ~-r+l (Which is not on
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, and let the row vectors of
19
~-r)
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has the same tangent space as A.
Theorem (7.4).
Given a non-degenerate Hermitian variety V _ the
Nl
tangent space at a point C of V _ intersects V _ in a degenerate Hermitian
Nl
Nl
variety V _2 of rank N-l contained in ~-l. The singular space of V _
N
N2
consists of the single point C.
Let the equation of VN..lbe
to
V~_l
~-1.
~T~(q)=o.
Let
~-l
be the tangent space
Let FO = C, F 1, ••• , FN_l be N independent points in
We can find a non-singular linear transformation if. = A ~, such
at C.
that Fo,F l' ••• , FN_1 become the fundamental points of the reference system.
Then the equation of
~
becomes
XN = 0
, and the equation of VN_l becomes
G y(q) = 0 , where G = «g'j»' (i,j = 0,1, ••• , N), is Hermitian and
~-l
-
~
of rank N+1.
Since F
we l1ave
= 0,
goj
j
O
is self. conjugate and is conjugate to F ,F , ••• , F _ ,
l 2
Nl
= 0,1,
••• , N-l, and gio
= 0,
can therefore write
'0 . • 0 0
•••
• • • •• • • • • •
0
G
.a
=
0
= 0,1,
i
••• N-l.
We
•
. . • ...
• gON .
• • •
• gill
•
.~
G**
••• •
•
t
••
•
.a
.~-i,N
•
• • • • • • • • • • • • • •• • • • •
~O·
•
where G**
= «gij»'
i,j.
= 1,2,
since G is non-singular.
Also
det
G=
~-l
••• , N-l.
-gON~O
It follows that det G**
Regarding
em ~
.
~s
.
~-l,N·~
.
,
Now gON and gNO are non-null
det G** •
non-null so that the rank of G** is N-l.
as a projective space of N-l dimensions, we have seen that
20
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*T
* -M(q)= 0,
the equation of the section VN_2 is :l G:l
*T
= (YO,Yl ,
... , Y - )
Nl
and G* is the Hermitian matrix obtained by retaining only the first N rows
and columns
0f
G.
* 1.S
.
•
S J.nce
G
tiona! null row and colwnn, rank
The row vector of C
~-l
~
where:l
** except
G* = rank G** = N-l.
the same as G
that it has an addi-
F0 when regarded as a point of projective spac.,
Since ..i-,,:/G*:: 0, C is a singular point of V _ •
Nl
Since the rank of VN_ is N-l, the singular space has d.i.mension 0, and must
2
is ...$..)*T = (1,0, ... 0).
therefore consist of the single point C.
Corollary.
Let
V _ be a degenerate Hermitian variety of rank
Nl
r < :N+l, with singular space
at a regular point C.
Then
~-r.
Let
~-l
be the tangent space to VN_J.
~-l
intersects V _l in a Hermitian variety
N
V _ of N-2 dimensions and rank r-l, whose singular space is the (N-r+l)N2
flat ~-r+l containing C and ~.-r •
8.
Number of points on a Hermitian variety.
Let
SN+l (q2) denote
the vector space of row vectors of order N+l, with elements from GF(q2), and
let ~+l(q) have a similar meaning with relation to GF(qj.
x
T
T
y"
=
To any vector
(xo,~, ••• , XN) belonging to SN+l (cf) let there correspond a vector
= (YO,Yl ,
... , YN) belonging to SN+l(q) where Yi
= X{+l
, (i = 0,1, ••• , NI"
It follows from the correspondence between the elements of' GF(q2) and GF(q)
discussed in section ?, that to each ~T there corresponds a unique y"
T
, but
to each y"T with r non-zero coordinates there correspond (q+l)r vectors !T
belonging to
SN+l(q), each vnth r non-zero coordinates.
Now let
X
zero coordinates.
be any point of PG(N, q2) with rovT vector
Then any one of the q2_1 row vectors
'vill represent X, where
P
p
~T having r non-
~T of ~+l (q2)
is any arbitrary non-zero element of GF(q2).
Let y"T be the row vector of SN+l (q) which correspondsto !,T , and let Y be
21
the point of PG(N, q) with rOi., vector r
X.
The point Y of PG(N,q) is given
T
•
We then say that Y corresponds to
uniquely by the point X of PG(N,q) for
if ire take p !T as the vector representing X, then the corresponding vector
of SN+l(q) is
arT where a
= pq+l
, and represents the same point of PG(N,q)
T
Conversely let iL be a vector of SN+1(q) representing a point Y
as -;,rT •
of PG(N,q).
If r
T
has r non-zero coordinates then we get (q+l)r distinct
vectors of SN+1 (q2) corresponding to yT.
one of the q-l row vectors
zero element of GF(q).
T
aiL of SN+l(q), where 'a' is any arbitrary non-
To each of these vectors there correspond (q+l)r
vectors of SN+l (q2).
Thus to the
all the non-zero elements of GF(q»
SN+l (q2).
Now Y can be represented by any
q-l
vectors ai/.T (where 'a' ranges over
there correspond (q_l)(q+l)r vectors of
But any q2_1 of these vectors which differ merely by a multiple
of some non-zero element p of GF(q2) represent the same point of PG(N, q2).
Hence to each point of PG(N,q) with r non-zero coordinates tl'ere correspond
(q_l)(q+l)r/(q2-l)or (q+l)r-l points of PG(N,q2).
Now let
V _
Nl
be a non-degenerate Hermitian variety in pG(N,q2).
By a suitable choice of the frame of reference we take its equation in the
N
canonical form
(8.1)
Let
E
i=O
x.x. = 0
or
J. J.
q+l
X
o
q+l
q+l
+:x:t
+ ... +XN
=0
E be the hyperplane of PG(N,q) with equation
(8.2)
In the correspondence between the points of PG(N,q) and pG(N,q2) just
described, if X lies on V _ then Y lies on E and conversely. Let
Nl
n = (qr+l_l)/(q_l) denote the number of points on an r-nat in PG(N,q).
r
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The number of points on
(d.3 )
~
which have exactly r non-zero coordinates is
r) no]
(n+l)
r [nr _2 - (r)
1 nr _3 + (r)
2 nr -4 + ••• + ( -1) r-2 (r-2
Hence the total number of points on V _ is
N1
N+l
E (N+l)[(q_l)r-l_ (_l)r-l]/q
r=l r
=
We have thus, proved:
Theorem (8.1).
The number of points on a non-degenerate Hermitian
Ie
Corollary.
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The number of points on a degenerate Hermitian variety
V _ of rank r < N+l
Nl
(8.5)
in PG(N,q2) is
(q2_1)f(N_r,q2) ~ (r-l,q2) + f(N-r,q2) + ~(r_l,q2) ,
where <I>(N,q2) is given by (8.4),
f(k,q2)
= [q2(k+l)_
l]/(q2_ 1 ) •
Let
L..N-r be the si.ngular space of VN_l then the nwnber of points
in ~- is f(N-r,q2). Also let E 1 be an (r-l)-flat disjoint from ~•
~-r
r~-r
Then from theorem (7.1), Er _l intersects V _l in a non-degenerate Hermitian
N
variety Vr- 2 contained in Er- l '
The number of points on Vr- 2 is
~(r_l,q2). Now from the corollary to theorem (7.2) every point of V _
Nl
belongs to some line joining a point of L..._
with a point of V 2' Two
iN-r
rsuc.h lines cannot have a p6int in common outs ide of V 2 or L..._
• Suppose
riN-r
if
pos~i.ble
~
~-r
not in
or Vr-2.
Then Al snd
be in
~-r
and the points B and B in
l
2
If possible let the lines "\B and A B intersect in P, a point
l
2 2
Vr _2 •
the points "\ and
~
are distinct.
If not they would coin-
cide vdth the point of intersection of the two lines vn1ich would mean that
P lies in Lw-r.
and
~B2
Similarly Bl and B2 are distinct.
lie in the plane
P~~.
Hence the lines
However both AlB
~~
in a point Q, which therefore lln.1st be connnon to.E..-rr-r
and BI B2
and
E l'
r-
l
intersect
This
contradicts the fact that Eland .E.._
are disjoint. Each line joining
r-rr-r
and V 2 contains q2_l points not contained in either
a point of .E.._
-rr-r
r~T-r and
on
Vr _2 •
Hence V _ contains
Nl
.E.._
or V 2'
fl-r
r-
9.
f(N-r,q2)$ (r_l,q2)(q2_l ) points not
This proves the corollary.
Flat spaces contained in Hermitian varieties.
(9.1).
conjugate with respect to V _ •
Nl
~T
Let the equation of V _ be
Nl
H !(q)= 0, where
H is Hermitian
Let c T and d T be the row vectors of C and D.
Then
The row vector of any point A lying on the line CD can be written
as a
T
=
T
T
"1£ + 1~
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The line joining two points C and D on a Hermitian
variety V _ is completely contained in V _ if and only if C and Dare
Nl
Nl
or order N+l.
-.
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We shall first prove the following lennna.
Lemma
I
I
= ( " 1 c:.. +
"2~J
T
•
V _ we lln.1st have
N1
If
CD is completely contained in
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Hence from
(9.1)
J Jq cT H d(q) +
12This implies that
suppose £T H £(q)
E? H !!(q)
°'.
=h ~
=°
J: Jq dT H c(q)
21--
= 0,
Le.
Then from lemma
element t.. of GF(q2) such that h~ + lit.. 1= 0.
then
h~ +
q
J1J2
lit..
= -t..,
= 0,
q
J2 J l
C and D are conjugate.
T-_
= t..
,£Il£
(q) -
=h
(3.1)
= 0,
we can find a non-zero
Now let us choose J
l
=1
I
J
2
=
so that from (9.3) we have
This is a contradiction.
Then cT H d(q)
Conversely suppose C and D are conjugate.
!!T H £(q)
If not
so that
(9.2)
is satisfied.
=
° and
Hence every point of the line
CD is on V _ •
N l
~orollary.
The necessary and sufficient condition for any t-flat
to be completely contained in V _ is that any two points of
N l
respect to V _ •
Nl
If
~t
~t
are conjugate "ri th
is contained in V _ ' and a point C of
Nl
"'egular point of V _ then
Nl
~t
~t
~t
is a
is contained in the tangent space to V _ at
Nl
C.
The first part is obvious.
is any point of
~
For the second part we observe that if D
, then D is conjugate to C, and is therefore contained in
the polar hyperplane of C, 14hich in this case is the tangent space to V _
Nl
at C.
Theorem
(9.1).
If N
= 2t+l
or 2t+e, then a non-degenerate Hermitian
variety V _ contains flat spaces of dimensions t
Nl
and no higher.
He can without loss of generality take the equation of V in the
N
canonical form xTx(q)
N+l.
= 0,
1. e., we take H =
~+l
' the unit matrix of order
Suppose V contains a t-flat determined by the t+l independent points
N
UO' Ul , ••• , Ut with row vectors
~'~l'
••• ,
~t.
conjugate to each other with respect to V _ •
N l
25
Any two of these points are
Hence
I
I
= 0,1,
i,j
-.
••• , t •
Let
oo
u
u.IO
u
ll
• ••
•••
u
to
• ••
u
tl
• ••
U
uT =
(9.5)
U:w
• ••
·..
u
tN
be the (t+l) x (N+l) matrix whose row vectors are
T
Since the rows of U are independent its rank is
least t+l independent colunms.
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I
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I
I
uON
Ol
,
~,~i,
t+1.
..., ~ .
Hence we can find at
We can suppose the first t+l columns
of UT to be dependent, for if this is not true
merely by a permutation of coordinates.
'We
can achieve it
Now the equation (9.4) may be re-
written as
Let
and ~
rank
_I
ui
be the matrix consisting of the first
the matrix consisting of the last N-t columns.
(~) ~ N-t. Now from (9.6)
(9.7)
(9. 8)
:.
Since
ui ui + ~ u~ q)
=0
•
x -.. x(q) is an automorphism of GF(q2) ,
ranl~ U1T = rank U1q = t+1. Hence
t+l
columns of
T
Tllen rank (u )
l
if ,
= t+l
,
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I
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= rank (ll1 u(q»
1
t+l
= rank(
~
which
ShOvlS
that
N
Changing
t
~
-ui U~ q) )
N - t ,
2t+l •
to t+l we f'ind that if' V _ contains a f'lat space of'
N1
dilOOnsions t+l, then N
~
2t+3.
Hence if
N
= 2t+l
or 2t+2 then V _ cannot
Nl
contain a f'lat space of' dimensions higher than t.
We shall next
ShOVl
that if' N
= 2t+1
nru.tually conjugate points on V _ •
N1
o•
can always find t+l
\'1e
The f'lat space of'
mined by these points nru.st lie in V _ •
N l
!:N-1 be the polar space of' U
or 2t+2,
~-l
dimensiom deter-
Choose any point U on VN_l"
Let
o
~-l
Then
t
intersects V _ in a degenerate
Nl
Ie
Hermitian variety V -2 contained in
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I
I
disjoint f'rom U and intersecting V _ in a non-degenerate Hermitian variety
o
N 2
.e
I
N
(cf'. Theorem (7.4)).
of' which U
o
Now we can f'ind an (N-2)-flat ~-2 lying in ~-l '
V _ contained in
N3
~-2 (cf'. Theorem (7.1».
Since
~-l
U
1
lies in
is the singular space
Let
o•
it is conjugate to U
U
l
Now let
space to V _ when considered as a variety of the space
N3
int.ersect V _
N
space.
3
be any point on V _
N 3
~-3
~-2'
•
be the tangent
and let it
in the Hermitian variety V -4 of which U is the singular
N
1
Again in
LN-3
we can f'ind a f'lat space ~-4 of' dimension N-4 disjoint
f'rom U and intersecting V -4 in a non-degenerate Hermitian variety V _ conN5
N
l
tained in
~-4.
Let U be in VN- •
2
5
Then U2 is conjugate to both U1 and Uo•
Continuing in this way '\ole get points U ' U ' ••• , Ut nru.tually conjugate to
O l
one another, U lying on V - - •
t
N 2t 1
to carry this process f'urther.
If'
N
= 2t+l
The f'lat space
U ' U ' ... , Ut lies completely in VN_l •
o l
27
or 2t+2, vle will not be able
!:t
determined by
Let
teN, t, q2) denote the number of t-flats contained in a non-degenerate
Hermitian variety V _ in PG(N, q2).
Nl
(~.lO)
1jr(N,O,q2)
=
Then we know that
4>(N,q2); 1jr(2t+l,k,q2)
for k > t,
t(2t+2,k,cf)= 0
<\leN, q2) is given by (8.4).
where
We shall next calcu1.ate the value of
First suppose N = 2t+1.
tangent space
tained in
~t-l
=
~t
at
C to V2t
Let
teN, t, q2) when N
C be any poin t on V2t.
~t
or 2t+2 •
Then the
cuts it in a Hermitian variety V _ con2t l
~t' for which C is the singular space.
contained in
= 2t+l
He can find a (2t-l) -flat
and disjoint trom C intersecting V2t _ in a non-degenerate
l
Hermitian variety V - •
2t 2
Now V _ contains
2t 2
t(2t-l,t-l,q2) (t-l)-flats.
Any of these (t-l)-flatstogether with C determines a t-flat contained in VN_l"
Conversely if E is a t-flat contained in V _ and passing through C, then it
t
Nl
~t and intersects
is contained in
~t-l in a (t-l)-flat contained in V
_ •
2t 2
Hence the number of t-fla.ts contained in V _ and passing through a fixed
Nl
point C is
t(2t-l,t-l,qO:;).
'qut the number of points on V is cI>(2t+l,q2),
2t
where ljl(N, q2) is given by (8.4). We thus get t(2t-l, t-l, q2) cI>(2t+l, q2)
t-f1ats by considering all points of V • Here each t-flat has been counted
2t
f(t,cf) = (q2(t+1)_ 1)/(q2_1) times since this is the number of points on a
t-flat.
Hence
t(2t+1,t,q2)
=
t(2t-1,t-1,~) cI>(2t+1,q2)
f(t, q2)
=
(q2t+l+ l)t (2t-1,t-1,q2) •
By successive reduction
2
1jr(2t,t,q )
since
t{1,O,q2)
= q+1
= (q2t+1+
l)(q2t-l + 1) ••• ( q+1),
•
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In the same manner we can prove that
1f(2tiQ, t, q2)
Theorem (9.2).
If
= (q2tiQ+l )(q2t+l )
teN, t, q2)
... (q3+l ) •
denotes the number of t-fiats on a non-
degenerate Hermitian variety V _ in PG(N,q2), then
Nl
1f(2t+l, t, q2)
=
1f(2t+2, t, q2)
= (q2t+3+l )(q2t+l+l )
(It+l+l )(q2t-l+l ) ••• (q+l) ,
•••
Some designs associated with a non-degenerate Hennitian variet;r
10.
of 2 dimensions ip PG(3,q2).
Let
V
2
be a non-degenerate Hermitian variety in PG(3,cr).
It follows from theorem (8.1), that V contains (q3+l )(q2+l) points.
2
The case
q
=2
vlith 45 points.
is of special interest.
In this case V is a cubic surface
2
Again from theorerns(9.l) and (9.2), V,?- does not contain
any plane but is ruled by lines, (q3+l )(q+l) in number.
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q
=2
the number of lines is 27.
In the special case
The lines lying on V2 vTill be called
generators of V •
2
From theorem (7.4), the tangent plane to V2 at any point C, intersects
V in a degenerate Hermitian variety V of rank 2, with C as a singular
l
2
point.
It follows from. theorem (7.1), that if we take a line t
tangent plane at C, disjoint from C, then t
v~uld
in the
intersect V in a nonl
degenerate Hermitian variet;jr V of dimension 0, contained in t.
o
sh~m
It was
in section 5 that V consists of a set of q+l distinct points
PO,P , ... , P •
q
l
o
It novT follows from the corollary to theorem (7.2) that
V1 consists of the set of q+1 concurrent lines CPO' CPl' ••• , CPq.
Thus
the tangent plane to V at any point C, meets V in a set of q+l generators
2
2
passing through C.
Conversely, from the corollary to Lemma. (9.4), any
29
generator through C is contained in the tangent plane at C.
We have thus
shown: Through any point C of V , there pass exactly q+l generators which
2
constitute the intersection withV2 of
th~tangent
plane at C.
Now through
C there pass q2+l lines lying in the tangent plane at C, out of which q+l
are generators.
The remaining cf-q lines through C, which lie in the tangent
plane meet V only in the single point.
2
Lines meeting V in a single point
2
C 'Hill be called tangents to V at the point where they :meet V •
2
2
C there will pass q4 lines not lying in the tangent planp. at C.
(q+l) generators through C contain q3+q2+l points of V •
2
Now the
Hence there are
2
q5 points of V not lying on the tangent plane at C.
Through
On the other hand any
line through C not lying in the tangent plane must meet V2 in a Henni tian
o* '
variety V
which must consist of either q+l points or a single point,
according as the rank is 2 or 1.
Since each of the q5 points of V2' not con-
tained in the tangent plane must
lie on some line through C, each of the
q4 lines passing through C and not contained in the tangent plAne at C must
~xactly
intersect V
2
must be of rank 2.
secants.
'1+1 points, one of which is C.
*o
This shows that V
Lines intersecting V2 in '1+1 pom ts may be called
Any arbitrary line not a generator of V must either be a tangent
2
or a secant.
Let
l
be any generator of V2.
to V at two distinct points of l
From theorem (7.3), the tangents
must be distinct.
2
There are exactly
'12 + 1 points on l , and exactly cf+l planes pass through
plane tl1rough a generator is tangent to V at some point,
2
V in a set of q+l generators through the point of
2
point on V disjoint from a C;iven c;enerator l .
2
ing P and
l
30
~nd
contac~.
intersects
Let
Then the "plane
must be tangent to V at some point C on l .
2
Hence!&
l.
P
1!
be a
contain-
Since P is on
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the intersection of V and :rr,
2
CP must be a generator of V • Since:rr can
2
be tangent to V at only one point on CP so 1f is not the tangent plane at
2
P. Let :rr* be the tangent plane to V at p. Then the q+l generators of V
2
through P lie on :rr*
•e
I
Thus:rr
and
:rr* intersect in a single generator CPo
We have thus shown that given a generator
I,
of V and a point P of V not on
2
2
1,1 there passes through P exactly one generator Which meets I, in a point.
The concept of a partial
the authors in (4).
~eometry
(r,k,t) was introduced by one of
It is a system of two kinds of undefined elements called
'points' and 'lines' and an undefined relation of 'incidence' satisfying the
following axioms.
I_
I
I
I
I
I
I
I
2
Al.
Any two points are incident with not more than one line.
A2.
Each point is incident with r lines.
A3.
Each line is incident with k points.
A4.
If the point P is not incident with the line I, there are
exactly t lines (t ~ 1) tlLrough P intersecting
Theorem (10.1).
1,.
The configuration of points and generators of a
Hermitian variety V in PG(3, q2) form a partial geometry (q+l, q2+1, 1).
2
All the axioms Al-A4 are satisfied in view of the results already
proved.
From the connection established between partial geometries and
partially balanced incomplete block (PBIB) designs in
(4),
it follows that
by identifying the points of V with treatments, and the generators of V
2
2
wi th blocks we get the PBIB design with parameters
(10.1)
v = (q2+1 ){q3+1), b = (q+l){q3+1 ), r = q+1,
n
l
= q2(q+l), n = q5, ph = q2_1,
2
31
P~l
k = q2+1,
= q+l, \
= 1, ~ = 0 •
This design was otherwise obtained by Ray Chaudhuri
(14).
I
I
The case
q = 2 was obtained earlier by Clatworthy and one of the authors (6).
For the
definition and other properties of PBIB designs the reader is referred to
(5, 7, 8, 9).
Let
ator.
Co
and
C
l
be two distinct points of V not on the same gener2
secant to V2 and intersectsV in
3
11:
1
tl
Denote the line joining Co and C by
l
•
Then, ,t must be a
l
q-l other points C ' ••• , C •
2
.
q
be the tangent planes to V at Co and C respectively.
2
l
pass through C •
l
"lOuld make COC
l
otherwise C would be on the section of V by
l
2
a generator contrary to the hypothesis.
not pass through CO.
Then
Now
Let
Let
1t
O
1t
cannot
11:
and this
O
0
Similarly
'1.
can-
t 2 be the line of intersection of rco and
t 2 must be skew to t l • Since t 2 is a line on
1t
O
and
1t
l
•
disjoint with Co'
t 2 meets V2 in q+l distinct points DO,Dl , ••• , Dq • Now Di (i=O,l, ••• ,q)
is conjugate to both Co and C •
l
Hence the tangent plane Zi at D passes
i
through Co and C and so through the line
l
(i,j
= 0,1,
t
l
Thus C and D are conjugate
j
i
•
... , q-l) and the lines CiD j are generators of V •
2
incidentally shown that if t"lO points Co and C
l
~
We have
V do not lie on a
2
generator then there are exactly q+l points D on V such that both DiC
i
O
3
~
DiC
l
are generators of V2 •
Now consider the special case
q:::;2.
Then V is the cubic surface
2
It has 45 points and 27 generators.
Through each point there pass
To each point P of V
2
3 generators and on each generator lie 5 points.
we may associate a set of 12 points viz the points
(other than p) lying on the three generators through P. This set of points
will be called the block corresponding to P.
There are exactly 45 blocks,
and each point P of V belongs to 12 blocks viz the blocks corresponding to
2
32
-.
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the
]2
points (other than p) lying on the three generators through P.
Given two distinct points P and Q on V we shall show that there are
2
exactly three blocks which contain both P and Q. We have to consider two
separate cases.
Firstly let P and Q lie on a generator J,*.
Now
J,* contains
three other points besides P and Q, and both P and Q belong to the blocks
corresponding to each of these points.
secant.
Again suppose P and Q lie on a
Then from what has been shown above the line of intersection of the
tangent planes at P and Q meets V in
2
DiP and DiQ (i
= 0,1,2)
q+l
=3
are both generators.
points D ,D ,D such that
O l 2
Hence both P and Q belong to
the blocks corresponding to DO' D and D2 •
l
Now a balanced incomplete block (BIB) design is a set of
or treatments, arranged into b
sets or blocks such that (i)
contains k distinct treatments (ii)
v
objects
Each block
Each treatment appears in exactly r
blocks (iii) Any pair of objects occurs in exactly A blocks.
v, b,r,k, A are called the parameters of the BIB design (2,1l).
The numbers
We may thus
state:
Theorem (10.2).
If V2
is a non-degenerate Herm1 tian variety in
PG(3,~), and if the points of V
are identified with treatments,and if
2
corresponding to each point P on V we define a block consisting of all
2
points (other than p) on the generators through P, then the treatments and
the blocks form a BIB design with parameters
v = b = 45,
r
= k = ]2,
A=3 •
This design has been otherwise obtained by Shrikhande and Singh (16)
and by Takeuchi (17).
There are many other interesting balanced and partially balanced
33
incomplete block designs associated with Hermitian varieties.
These shall
be discussed in a separate communication.
References
1.
R. Baer. Linear algebra and projective geometry (Ne"VT York: Academic
Prpl;s, 1952).
2.
R. C. Bose. On the construction of balanced incomplete block designs.
Ann. Eugen. Lenton, 9 (1939), 358-399.
3.
On the application of finite projective geometry for
deriving a certain series of balanced Kirkman arrangements.
Golden jubilee commem. vol., Cal. Math. Soc., (1958-59), 341-354.
4.
Strongly regular graphs, partial geometries and
balanced designs. Pacific. J. Math., 13 (1963), 389-419.
5.
Combinatorial properties of parially balanced designs
and association schemes. Sankhya, series A, 25 (1963), 109-136
and Contributions to statistics, presented to Professor P. C.
Ma.ha.lanobis on the occasion of his 70th birthday (Calcutta:
Statistical Publishing Soc., 1964), 21-48.
partia1~
6.
R. C. Bose and W. H. C1atvTOrthy. Some classes of partially balanced
designs. Ann. Math. Stat., 26 (1955), 212-232.
7.
R. C. Bose and D. M. Mesner. On linear associative algebras corresponding to association schemes of partially balanced designs. ~. Math.
Stat., 30 (1959), 2l-~Q,
8.
~.
9.
R. C. Bose and T. Shimamoto. ~lassification and analysis of partially
balanced designs with two associate classes. J. Am. Stat. Ass., 47
(1952), 151-184.
C. Bose and K. R. Nair. Partially balanced incomplete block designs.
Sankhya, 4 (1939), 337-7)72.
10.
R. Carmichael. Introduction to the theory of groups of finite order
(New York: Dover Publications, 1956), Chapters XI and XII.
11.
H. B. Mann. Analysis and design of experiments (New York: Dover
Publications, 1949), chapter IX.
12.
E. J. F. Primrose. Quadrics in finite geometries.
Soc., 47 (1951), 299-304.
Proc. Carob. Phil.
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13.
D. K. Ray-Chaundhuri. Some results on quadrics in finite projective
geometries based on Galois fields. Can. J. Math., 14 (1962), 129-138.
14 ,
Application of the geoemetry of quadrics far
constructing PBIB designs. Ann. Math. Stat., 33 (1962), 1175 -1186.
15.
B. Segre. Lectures on modern geometrY (Rome: Edizioni Cremonese, 1960),
chapter 17.
16.
S. S. Shrikhande and N. K. Singh. On a method of constructing symmetrical balanced incomplete block designs. Sankhya, series A, 24 (1962),
25-32.
17.
K. Takeuchi. On the construction of a series of BIB designs.
Appl. Res., JUSE, 10 (1963), 48.
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stat.