the 3-dimensional additive group
Shigeru MUKAI
An m-dimensional linear representation of a group induces an action
on the polynomial ringC[z
1
;:::;z
m
] of m variables. This is called a linear
actiononthepolynomialring. In1890, Hilbert[2]showedthattheinvariant
ringwas nitelygenerated for classical representations of the special linear
groups. The following is known as his fourteenth problem:
Problem 1 Is the invariant ring C[z
1
;:::;z
m ]
G
of a linear action of an
algebraic group G nitely generated?
TheanswerisaÆrmativefortheadditivealgebraicgroupG
a
(Weitzenbock
[11], [9]). In 1958, Nagata[5] considered the standard unipotent linear ac-
tion
(t
1
;:::;t
n
) 2 C n
y C[x
1
;:::;x
n
;y
1
;:::;y
n
] =: S (1)
x
i 7!x
i
y
i 7!y
i +t
i x
i
; 1 i n;
of C n
on the polynomial ringS of 2n variables andshowed that the invari-
ant ring S G
with respect to a general linear subspace G C n
of codimen-
sion 3 was not nitely generated for n = 16. In this article, we shall prove
the following:
Theorem The invariant ring S G
of (1) with respect to a general linear
subspace G C n
of codimension r is not nitely generated if
1
2 +
1
r +
1
n r
1: (2)
In other words, S G
is not nitely generated if dimG = s 3 and if
n s 2
=(s 2). So the answer to Problem 1 is negative for G 3
a
. But the
following part is still open:
Supportedin partbytheJSPSGrant-in-AidforScienticResearch(A)(2)10304001.
Problem 2 Is the invariant ring C[z
1
;:::;z
m
] of a linear action of the
2-dimensional additive group G= G
a
G
a
nitely generated?
See Roberts [8] for non-linear actions.
Our proof of the theorem is based on the fact that the invariant ring
S G
is a certain Rees algebra (x1). In geometric term, the Rees algebra
is isomorphic to the total coordinate ring T C(X) of the blow-up X of the
projective space P r 1
at n points (x2). This ring TC(X) is graded by the
Picard group PicX ' Z n+1
and its support is EX, the semi-group of
eective classes on X. Hence TC(X) is not nitely generated if EX is
not so as semi-group (Lemma 2).
The simplest case is
G = (
(t
1
;:::;t
9 )
9
X
i=1 t
i
= 9
X
i=1 }(c
i )t
i
= 9
X
i=1 }
0
(c
i )t
i
= 0 )
C 9
; (3)
where }(z) is Weierstrass's }-function of an elliptic curve C = C=(Z+Z)
and c
1
;:::;c
9
are distinct points C. In this case, X is the blow-up of
P 2
at the nine points (1 : }(c
i ) : }
0
(c
i
)), 1 i 9. Assume that the
sum P
9
i=1 c
i
2 C is zero, for simplicity. Then the nine points are the
intersection of two cubics, X has an elliptic bration f : X ! P 1
and the
nine exceptional curves are sections of f. If the dierence c
i c
i+1 is of
inniteorderfor some1 i 8,thenthere areinnitelymanyexceptional
curves of the rst kind (cf. [6]). So S G
is not nitely generated. (Cf.
Remark 1 at the end of x4.)
The proof of the theorem (x4) is similar but we replace the elliptic
bration by the symmetry of PicX with respect to the Weyl group of the
Dynkin diagram T
2;r;n r
with n vertices (x3):
r
n r
(4)
which was introduced by Dolgachev[1]. As is well known the inequality
(2) is equivalent to the inniteness of the Weyl group of this diagram
(Lemma 4). If G C n
is general and if (2) is satised, then there exist
innitely many exceptional divisors on X. Therefore, EX and hence
TC(X) are not nitely generated (Lemma 3).
Let G C n
be a linear subspace of codimension r and
n
X
i=1 a
(1)
i t
i
= n
X
i=1 a
(2)
i t
i
= = n
X
i=1 a
(r)
i t
i
= 0 (5)
a system of dening equations. Since x
1
;:::;x
n
are G-invariant, we obtain
the induced action of G on the localization
S[x 1
1
;:::;x 1
n
] = C[x 1
1
;:::;x 1
n
;y
1
;:::;y
n
] = C[x 1
1
;:::;x 1
n
; y
1
x
1
;:::; y
n
x
n ]:
Since (t
1
;:::;t
n
) 2 G acts by the translation y
i
=x
i 7! y
i
=x
i +t
i
, the invari-
ant ring S[x 1
1
;:::;x 1
n ]
G
is generated by
n
X
i=1 a
(1)
i y
i
x
i
; n
X
i=1 a
(2)
i y
i
x
i
; :::; n
X
i=1 a
(r)
i y
i
x
i
(6)
over the Laurent polynomial ring C[x 1
1
;:::;x 1
n
]. Let
J (1)
(x;y); J (2)
(x;y); :::; J (r)
(x;y) 2 S G
(7)
be the products of (6) and the monomial Q
n
i=1 x
i
. Let V be the subspace
and R the subring of S G
generated by them. R is a polynomial ring and
V is its degree one part. The invariant ring S G
contains R [x
1
;:::;x
n ] and
S[x 1
1
;:::;x 1
n ]
G
coincides with R [x 1
1
;:::;x 1
n
]. Obviously we have
S G
= S[x 1
1
;:::;x 1
n ]
G
\S = R [x 1
1
;:::;x 1
n
]\S: (8)
Let V
1
be the linear subspace of V consisting of J(x;y) which do not
contain the monomial y
1 Q
n
i=2 x
i
. Then V
1
V is of codimension 1.
A polynomial J(x;y) 2 V is divisible by x
1
if and only if it belongs to
V
1
. Let I
1
R be the ideal generated by V
1
. Dene V
i
V and I
i
R
for 2 i n similarly. If F(x;y) 2 R belongs to the b
i
-th power I b
i
i ,
then F(x;y) is divisible by x b
i
i
and the quotient F(x;y)=x b
i
i
belongs to S G
.
Hence S G
contains
R [x
1
;:::;x
n ]+
X
b
1
;:::;b
n 0
(I b
1
1
\\I b
n
n )x
b
1
1
x b
n
n
R [x 1
1
;:::;x 1
n
] (9)
as its subring. The following was proved in [5] in the case of codimension
3.
Proposition The invariant ring S of the action (1) with respect to a
subspace G C n
coincides with the extended multi-Rees algebra (9) of
(R : I
1
;:::;I
n ).
Proof. It suÆces to show the following
claim : f(J (1)
(x;y);:::;J (r)
(x;y)) 2 R is divisible by x b
i
i
if and only if
f(J (1)
;:::;J (r)
) belongs to I b
i
i .
If a (1)
i
;:::;a (r)
i
are allzero, then J (1)
(x;y);:::;J (r)
(x;y) are alldivisible
by x
i
. The claimis obvious,since none is divisible by x 2
i
and since V
i
= V.
So assume the contrary. By reordering (7), we may assume that a (1)
i
6= 0.
Put
z
1
= J (1)
=a (1)
i
;z
2
= J (2)
a (2)
i z
1
;:::;z
r
= J (r)
a (r)
i z
1 :
Then
f(J (1)
;:::;J (r)
) = f(a (1)
z
1
;a (2)
z
1 +z
2
;:::;a (r)
z
1 +z
r )
and this belongs to the ideal (z
2
;:::;z
r )
b
i
if and only if f(J (1)
;:::;J (r)
)
belongs to I b
i
i
by the lemma below. When regarded as polynomials of
x
1
;:::;x
n
;y
1
;:::;y
n
, the r 1 polynomials z
2
;:::;z
r
are divisible by x
i
and only z
1
is not. Therefore, f belongs to (z
2
;:::;z
r )
b
i
if and only if
f(J (1)
(x;y);:::;J (r)
(x;y)) is divisible by x b
i
i .
Lemma 1 Let I be the ideal of C[z
1
;:::;z
r
] generated by linear forms
vanishing at
(a (1)
;a (2)
;:::;a (r)
) 2 C r
:
Assume that a (1)
6= 0. Then a polynomial f(z
1
;:::;z
r
) belongs to the b-th
power I b
if and only if
f(a (1)
z
1
;a (2)
z
1 +z
2
;:::;a (r)
z
1 +z
r )
belongs to the b-th power of the homogeneous ideal (z
2
;:::;z
r ).
For small values of r, the invariant ring is very explicit.
Example 1 (r = 1) Assume that G C n
is dened by P
m
i=1 t
i
= 0 for
1 m n. Then S G
is generated by x
1
;:::;x
n and
( y
1
x
1
++ y
m
x
m )
m
Y
i=1 x
i :
Example 2 (r = 2)AssumethatG C n
isdenedby n
i=1 t
i
= n
i=1 c
i t
i
=
0. Then c
i J
1
(x;y) J
2
(x;y) is divisibleby x
i
and the quotient (c
i J
1 (x;y)
J
2
(x;y))=x
i
belongs to S G
for every 1 i n. S G
is generated by these
invariants over C[x
1
;:::;x
n ] if c
1
;:::;c
n
are distinct.
2 Total coordinate ring
Forourpurpose, it ismoreconvenientto statethe propositioningeometric
term. LetP r 1
= ProjR be the (r 1)-dimensionalprojectivespace whose
homogeneous coordinates are (7). In the sequel we assume that
(}) r 3 and any two of n vectors (a (1)
i
;a (2)
i
;:::;a (r)
i
) 2 C r
;1 i n,
are linearly independent.
(The study of S G
for the action (1) is easily reduced to this case.) Then n
points
p
i
:= (a (1)
i : a
(2)
i
::::: a (r)
i
) 2 P r 1
; 1 i n; (10)
are well-dened and distinct. The ideal I
i
R is generated by the linear
forms vanishing at p
i . Let
: X = X
G
!P r 1
be the blow-up at these n points. The isomorphism class of X
G
does not
depend on the choice of the dening equation (5). The Picard group is a
free abelian group of rank n+1. The pull-back h of the hyperplane class
H and the classes e
i
, 1 i n, of the exceptional divisors form a basis,
which is called the standard basisof PicX
G
(withrespect to). The direct
sumof the spacesof global sections of all line bundles (up to isomorphism)
M
a;b
1
;:::;b
n 2Z
H 0
(X;O
X
(ah b
1 e
1
b
n e
n )) '
M
L2PicX H
0
(X;L) (11)
is a graded ring, which is called the total coordinate ring of X and denoted
by TC(X). In our case, TC(X
G
) is the Rees algebra (9), or more precisely,
it is the Z n
-graded ring (9) plus the extra grading of the polynomial ring
R . By the proposition, we have
Corollary Under the condition of (}), the invariant ring S G
of the action
(1) with respect to G C n
is the total coordinate ring TC(X
G
) of the
blow-up X
G .
Let A =
2 A
be an integral domain graded by a free abeliangroup
. The subset fjA
6= 0g of is a semi-group. This is called the support
of A and denoted by SuppA.
Lemma 2 If SuppA is not nitely generated as semi-group, neither is A
as a ring over A
0 .
Proof. Assume that A is nitely generated. Then nite nonzero homoge-
neous elements a
i 2 A
i
, 1 i N, generate A and
1
;:::;
N
generate
SuppA.
For example, the support of TC(X) as Z n+1
-graded ring is the semi-
group
EX := fL 2 PicX jH 0
(X;L) 6= 0g;
of linear equivalence classes of eective divisors on X. If EX is not
nitely generated as semi-group, neither is T C(X). The following is basic
for our analysis of EX.
Lemma 3 Let : X ! Y be the blowing up of a projective variety Y at
a point. Then the linear equivalence class of the exceptional divisor E of
belongs to any system of generators of the eective semi-group EX.
Proof. Assume that E is linearly equivalent to the sum D
1 + D
2
of two
eective divisors. Let H be the pull-back of an ample divisor on Y. Then
the intersection number (E:H m 1
), m = dimX, is zero. Hence so are
(D
1 :H
m 1
) and (D
2 :H
m 1
). Therefore, both SuppD
1
and SuppD
2 are
contained in E and either D
1 or D
2
is zero.
If X and X 0
are isomorphic in codimension one, then the Picard groups
are the same and EX = EX 0
. So we call D X a ( 1)-divisor if
there is a birational map f : X ! X 0
and a morphism : X 0
! Y
such that f is an isomorphism in codimension one, is the blowing up of
a projective variety Y at a smooth point and D is the strict transform of
the exceptional divisor of . By the lemma, the class of a ( 1)-divisor is
containedin any system of generators of EX. Hence EX is not nitely
generated if X has innitely many classes of ( 1)-divisors.
Letbethe latticeofrankn+1withorthogonalbasish;e
1
;:::;e
n
. In view
of the standard Cremona transformation (see the next section especially
the formula (16)), we set (h 2
) = r 2 and (e 2
i
) = 1 for 1 i n. For
= ah P
n
i=1 b
i e
i
2 , we denote its coeÆcient a in h by deg. We put
= rh P
(r 2) P
n
i=1 e
i
, which corresponds to the anti-canonialclass of
the blow-up of P r 1
at points. The orthogonal complement of together
with its basis
e
1 e
2
; e
2 e
3
; :::; e
n 1 e
n
and h r
X
i=1 e
i
(12)
becomes a root system. The Dynkin Diagram is (4), that is, T
2;r;n r with
three-legs of length 2;r and n r. For a subset I [n] := f1;2;:::;ng of
cardinality r,
I
= h P
i2I e
i
is a root. The reection R
I
with respect to
I
is as follows:
8
<
:
h 7! h+(r 2)
I
= (r 1)h (r 2) P
i2I e
i
e
i
7! e
i +
I
for i 2 I
e
j
7! e
j
for j 62 I
(13)
Let W be the Weyl group of (12). By denition, W leaves invariant,
that is, rw(h) (r 2) P
n
i=1 w(e
i
) = for every w 2 W. In particular, we
have
rdegw(h) (r 2) n
X
i=1
degw(e
i
) = r: (14)
Lemma 4 If the inequality (2) holds, then the W-orbit of e
n
is innite.
Proof. The assumption implies r 3. Let w be an element of the Weyl
group. There exists a subset I [n] of cardinality r such that
X
i2I
degw(e
i )
r
n n
X
i=1
degw(e
i ):
By (14) we have
degw(
I
) = degw(h)
X
i2I
degw(e
i
) degw(h)
r 2
n(r 2)
(degw(h) 1);
I
2)degw(
I
) is also positive. It follows that the degree is increased by
a suitable reection R
I
. Hence, the orbit W h is innite. So is W e
n by
the equality (14).
The Weyl group of T
p;q;r
is innite if and only if 1=p+ 1=q + 1=r 1
([3] Chap. 4). The lemma also follows from this.
LetC beanellipticcurveand
C
the(n + 1)-dimensionalvarietyPic r
C
C n
. This is canonically isomorphic to Pic r
C (Pic 1
C) n
. So the factor
permutation of C n
and the automorphism
(D;c
1
;:::;c
n
) 7!(D 0
;c 0
1
;:::;c 0
n );
8
<
: D
0
= (r 1) (r 2) P
r
i=1 c
i
c 0
i
= D c
1
c
i
c
r
for 1 i r
c 0
j
= c
j
for r+1 j n
dene the action of the Weyl group W on the variety
C
. For a real root
= ah P
n
i=1 b
i e
i 2
re
([3] Chap. 5), the reection R
interchanges
f
:
C
!Pic 0
C; (D;c
1
;:::;c
n
) 7!aD n
X
i=1 b
i c
i :
with f
. We denote the ber f 1
(0) by D( ).
Example 3 D(e
i e
j
), i 6= j, is the diagonal fc
i
= c
j
g. D(h
P
r
i=1 e
i )
consists of (D;c
1
;:::;c
n
) such that P
r
i=1 c
i
2 jDj.
The Weyl group W acts on the complement of all these bers:
C
[
2 re
D( ): (15)
4 Standard Cremona transformation
The map
: P r 1
! P r 1
; (x
1 : x
2
: : x
r ) 7!(
1
x
1 :
1
x
2
: : 1
x
r
); r 3;
is a birational transformation of the projective space P r 1
. It contracts
the r coordinate hyperplanes to the r coordinate points and its square
is called a standard Cremona transformation. Let P = fp
1
;:::;p
r
g and
Q = fq
1
;:::;q
r
g be a pair of sets of r points of P r 1
. If both P and Q
span P r 1
, then there exists the unique standard Cremona transforma-
tion which contracts the hyperplane H
i
passing through the r 1 points
p
1
;:::;p
i
;:::;p
r
to the point q
i
for every 1 i r. We denote this by
P;Q
. P and Q are called its center and cocenter, respectively.
P;Q
is the
rational map associated with j(r 1)H (r 2) P
n
i=1 p
i
j, the linear sys-
tem of hypersurfaces of degree (r 1) passing through P with multiplicity
r 2. (Thesum of r 1 of H
1
;:::;H
r
form abasis of the linearsystem.)
The indeterminacy locus of
P;Q
is the union I
P
:= [
1i<jr H
i
\H
j
of the
intersection of all pairs of the hyperplanes H
i 's.
Let X
P
and X
Q
be the blow-up of P r 1
with center P and Q, respec-
tively.
P;Q
inducesthe birationalmap
~
P;Q
fromX
P toX
Q
. The diagram
X
P
~
P;Q
! X
Q
# #
P r 1
!
P;Q
P r 1
iscommutativeand
~
P;Q
inducesan isomorphismbetween thecomplement
of the strict transformof I
P
and thatof I
Q
. Hence
~
P;Q
is an isomorphism
in codimension one. (More precisely,
~
P;Q : X
P
!X
Q
is the composite
of certain ops.) In particular it induces an isomorphism PicX
P
!
PicX
Q
between the Picard groups and that between the semi-groups of
eective classes. Let fh;e
1
;:::e
r
g be the standard basis of PicX
P
. Then
the standard basis of PicX
Q
consists of
(r 1)h (r 2) r
X
i=1 e
i
; and h e
1
e
i
p
r
; 1 i r: (16)
Proof of Theorem. Let C be an elliptic curve and take an (n+1)-tuple
(D;c
1
;:::;c
n
) from the W-invariant open subset (15)of
C
. The complete
linear system jDj embeds C into the (r 1)-dimensional projective space
P
D
:= P
H 0
(C;O
C
(D)). Let p
1
;:::;p
n 2 P
D
be the image of c
1
;:::;c
n
by the embedding
D
. Since (D;c
1
;:::;c
n
) does not belong to the divisor
D(e
i e
j
)
C
for any 1 i < j n, the n points p
1
;:::;p
n are
distinct. Moreover, since it does notbelongsto D(
I
) for any I [n] with
jIj = r, any r of p
1
;:::;p
n
spans the projective space P
D
(Example 3).
D
any r of p
1
;:::;p
n
as center. Put (D 0
;c 0
1
;:::;c 0
n
) = R
I (D;c
1
;:::;c
n
) and
p 0
i
=
D 0(c
0
i
) for 1 i n. Then we have the commutative diagram:
C = C
D
# #
D 0
P
D
!
I
P
D 0
where
I
is the standard Cremona transformationwhose center is fp
i ji 2
Ig and cocenter is fp 0
i
ji 2 Ig. Any point of C other than fp
i
ji 2 Ig does
not lie in the indeterminacy locus of
I
. Let : X !P
D
be the blowing
up at the n points p
1
;:::;p
n
and 0
: X ! P
D 0
at p 0
1
;:::;p 0
n
. Then
I
induces
~
I
between X and X 0
and we have the commutative diagram:
C = C
?
?
y
?
?
y
X
~
I
! X
0
# #
0
P
D
!
I
P
D 0
By our choice of (D;c
1
;:::;c
n
), the images p 0
1
;:::;p 0
n of c
1
;:::;c
n are
distinct and any subset of cardinality r spans P
D 0
. Hence we can perform
the standard Cremona transformation with any r of p 0
1
;:::;p 0
n
as center.
We cancontinuethisasmanytimesaswe like. Hencewe have thefollowing
by (13) and (16):
Lemma 5 If an (n+1)-tuple(D;c
1
;:::;c
n
) belongs to the open subset (15)
of
C
and if is in the orbit W e
n
, then there exists a ( 1)-divisor D
whose linear equivalence class is .
It is obvious that the same holds for the blow-up
~
X at p~
1
;:::;p~
n
if the
n-tuple (p~
1
;:::;p~
n
) 2 P r 1
P r 1
belongs to a neighborhood of
(p
1
;:::;p
n
) in the classical topology. Hence, by virtue of Lemma 4,
~
X
contains innitely many classes of ( 1)-divisors if (2) holds. Therefore,
S G
for a general G C n
is not nitely generated by Corollary and two
lemmas in x2.
consider the diagonal subring
S TG
:= R [x]+ X
b0 (I
b
1
\\I b
n )x
b
R [x 1
]; x = n
Y
i=1 x
i
;
of (9), which is isomorphic to
M
a;b2Z H
0
(X
G
;O
X
(ah b(e
1
++e
n
))); (17)
in the case where n = 9 and G C 9
is of codimension 3. They show
that this is not nitely generated if 3D
P
9
i=1 c
i
2 C is of innite order.
The innite generation of S G
follows from this easily. Note that S TG
becomes nitely generated if 3D
P
9
i=1 c
i
is torsion but still S G
is not
nitely generated if the dierences c
i c
j
are general. Note also that
= 3h P
9
i=1 e
i
2 corresponding to 3D P
9
i=1 c
i
is an imaginary root
of the aÆne root system
?
of type T
2;3;6 .
Remark 2 If (2) holds and if c
1
;:::;c
n
2 C are general, then the image
of the restriction map
S G
= TC(X
G
) ! TC(CjD;c
1
;:::;c
n ) :=
M
a;b
1
;:::;b
n 2Z
H 0
(C;O
C (aD
n
X
i=1 b
i c
i ))
is not nitely generated. This gives another proof of Theorem. The image
is similar to the bi-graded ring
M
m;n2Z H
0
(C;O
C
(mc+nd))
obtained from two points c;d2 C. If the dierence c d2 C is of innite
order, then the support is fm + n > 0g [ f(0;0)g, which is not nitely
generated as semi-group (cf. [7]).
References
[1] Dolgachev, I.: Weyl groupsand Cremonatransformations,Proc.Symp.
Pure Math. 40(1983), 283-294.
[2] Hilbert, D.: Uber die Theorie der algebraischen Formen, Math. Ann.,
36 (1890), 473-534.
[3] Kac,V.G.: Innite dimensionalLie algebras,2nd.ed.,CambridgeUniv.
Press., 1983.
[4] Mukai, S.: Moduli Theory I, II, Iwanami Shoten, 1998, 2000, Tokyo.
(English translation : An introduction to invariants and moduli, to
appear.)
[5] Nagata, M.: On the fourteenth problem of Hilbert, Int'l Cong. Math.,
Edingburgh, 1958.
[6] ||: On rational surfaces, II, Mem. Coll. Sci. Univ. Kyoto. Ser. A,
33(1960), 271-293.
[7] Rees, D.: On a problem of Zariski, Illinois J. Math. 2(1958), 145-149.
[8] Roberts, P.: An innitelygenerated symbolic blow-up in apower series
ring and a new counterexample to Hilbert's 14th problem, J. Algebra,
132(1990), 461-473.
[9] Seshadri, C.S.: On a theorem of Weitzenbock in invariant theory, J.
Math. Kyoto Univ., 1(1962), 403-409.
[10] Steinberg, R.: Nagata's example, in `Algebraic Groups and Lie
Groups',Austral.Math.Soc.Lect.Ser.9, CambridgeUniv.Press,1997,
pp. 375{384.
[11] Weitzenbock, R.:
Uber die Invarianten von Linearen Gruppen, Acta.
Math., 58(1932), 230-250.
Research Institute for Mathematical Sciences
Kyoto University
Kyoto 606-8502, Japan
e-mail address : [email protected]
