NOTES ON COX RINGS
JOAQUÍN MORAGA
Abstract. These are introductory notes to Cox Rings. The aim of this notes
is to give a self-contained explanation of a talk given by the author in the
students seminar of algebraic geometry of the University of Utah.
Contents
1. Introduction and Motivation
2. Projective Varieties
3. Cox Rings
References
1
3
7
10
1. Introduction and Motivation
In 1902, David Hilbert propose 23 problems to the mathematical comunity to
measure the growth of mathematics during the early years of the new century.
In this oportunity, we will focus in the 14th Hilbert problem, whose solution had
important consequences in the theory of invariants, geometric invariant theory and
algebraic geometry in general.
In what follows, we will denote by k an uncountable algebraically closed field of
characteristic 0. For example, you can consider k = C. The original statement of
the Hilbert 14th problem is the following:
Problem 1.1. Let k(x1 , . . . , xn ) be the field of rational functions on n variables
over the field k and K ⊂ k(x1 , . . . , xn ) a subfield. Is K ∩ k[x1 , . . . , xn ] finitely
generated over k?
In what follows, we will be interested in the action of groups in rings and in
trying to understand the ring of invariants (i.e. the ring generated by all the
elements which are invariant by the action of the group). We will be particularly
interested in linear algebraic groups, which are groups isomorphic to subgroups of
the general linear group over k.
With the advances of David Mumford in theory of invariants, several mathematicians get interested in the following version of Hilbert’s problem:
Problem 1.2. Given a ring R finitely generated over k and a algebraic linear group
G acting on R. Is the algebra of invariants RG a finitely generated ring?
Remark 1.3. The Problem 1.2 can be deduced by Problem 1.1 as follows: If
we consider a linear algebraic group G acting over k[x1 , . . . , xn ], we can consider
the ring K = k(x1 , . . . , xn )G , and then deduce the Problem 1.2 with the case
R = k[x1 , . . . , xn ].
1
2
J. MORAGA
The following definition is given to introduce Theorem 1.7, which is a first attempt to give a positive answer to Problem 1.2.
Definition 1.4. Given a linear algebraic group G, we will call radical the component of the identity of the maximal normal resoluble subgroup of G. An element
g ∈ G is called unipotent if g − In is nilpotent. The unipotent radical of G is the set
of all the unipotent elements of the radical of G. We will say that G is reductive
if the unipotent radical is trivial, in other words, if the radical does not contain
unipotent elements.
Example 1.5.
• The product of two reductive groups is reductive. Moreover, the one-dimensional torus k ∗ is reductive, then all the n-dimensional
torus (k ∗ )n is reductive as well.
• The general linear group is reductive.
• The additive group (k n , +) is not reductive.
Proposition 1.6. (Main property of reductive groups) Given a reductivo group G
acting on a finite-dimensional k-vector space V , and a subspace H ⊂ V which is
fixed by the action of G (i.e. g · H ⊂ H for all g ∈ G). Then there exists a subspace
W 0 which is also fixed by G and W ⊕ W 0 = H.
Theorem 1.7. (Hilbert-Mumford) Let G be a reductive group acting on a ring R
finitely generated over k. The algebra of invariants RG is finitely generated over k
as well.
Proof. We will prove the case R = k[x1 , . . . , xn ]. Observe that we have a Z≥0 grading on both rings R and RG . We will denote
X
X
R=
Rn ,
RG =
RnG .
n∈Z≥0
n∈Z≥0
Since G is reductive, we can use Proposition 1.6 to write a decomposition Rn =
RnG ⊕ Rn0 for each n ∈ Z≥0 , as vector spaces. Then, for each n we have a projection
morphism ρn : Rn → RnG . The application ρ = ⊕n∈Z≥0 is a well defined morphism
ρ : R → RG , with the following formal property
(1.1)
∀g ∈ RG , f ∈ R,
ρ(gf ) = gρ(f ).
Consider the ideal I = ⊕n∈Z≥0 RnG .
there exists elements f1 , . . . , fk ∈ RG
Using Hilbert Basis Theorem we have that
such that I = hf1 , . . . , fk i, We will prove that
S, the subring generated by f1 , . . . , fk is the ring of invariants of R via the action
G. Clearly, the inclusion S ⊂ RG holds. On the other hand, pick h ∈ RG . We can
assume by induction that all the elements of RG which have lower degree than h
Pk
are in S. Observe that we can write h = i=1 hi fi , for hi ∈ R, given that h ∈ I.
Using property 1.1, we conclude the following
!
k
k
X
X
h = ρ(h) = ρ
hi fi =
ρ(hi )fi .
i=1
i=1
Since ρ(hi ) are G-invariant elements of degree lower than h, by the induction hypothesis we conclude that ρ(hi ) ∈ S. Thus, h ∈ S, we conclude the claim.
For a complete proof of the above Theorem, you can see [3]. The following, is
an example that shows that the Hilbert problem has a negative answer in general.
We won’t prove the details of the following construction, but we will return to this
NOTES ON COX RINGS
3
example in next sections. Since we already know that reductive groups does not
give a counter-example of the 14th problem, now we consider the action of the
additive group (k n , +).
Example 1.8. (Nagata Counterexample) Consider the ring R = k[x1 , . . . , xn , y1 , . . . , yn ]
with the action of k n given by
xi 7→ xi ,
i ∈ {1, . . . , n},
yi 7→ yi + ti xi ,
i ∈ {1, . . . , n},
where (t1 , . . . , tn ) denote the coordinates of k n . Consider
the subspace G = k n−r ⊂
P
n
k , of codimension r, defined by the equations
j ai,j tj = 0, with 1 ≤ i ≤ r.
Moreover, we assume that the matrix A = (ai,j ) does not have null entries and has
maximal rank. The action of G over the ring R is known as Nagata action. Given
that the elements x1 , . . . , xn are invariants by the action of G in R, we can consider
−1
the induced action of G in the ring R[x−1
1 , . . . , xn ]. Observe the following equality
y1
yn
−1
−1
−1
−1
R[x1 , . . . , xn ] = k[x1 , . . . , xn ]
,...,
.
x1
xn
P ai,j yj
−1 G
and
The ring of invariants R[x−1
1 , . . . , xn ] is generated by the elements
j xj
moreover we have the following equality
−1 G
RG = R ∩ R[x−1
1 , . . . , xn ] .
For each i ∈ {1, . . . , n} we will use the following notation


X ai,j yj
,
wi = (x1 · · · · · xn ) 
x
j
j
for i ∈ {1, . . . , r}. Observe that the algebra k[w1 , . . . , wr ] is contained in the ring of
invariants RG , since all its generators are invariant elements. Finally, the ring RG is
f
generated by all the elements of the form m
, where f is a homogeneous polynomial
f
is a
in the ring k[w1 , . . . , wr ] and m is a monomial in k[x1 , . . . , xn ] such that m
polynomial. For n and r big, those elements are infinitely many.
2. Projective Varieties
In what follows, we will assume that our varieties are projectives. A projective
algebraic variety is the zero locus, of a homogeneous prime ideal in k[x1 , . . . , xn ],
in Pn . Recall that the zero locus of such polynomials does not depend on the
representative that we take of a point of Pn . In other words, given a homogeneous
prime ideal I ⊂ k[x0 , . . . , xn ], we have an associated variety
X(I) = {p ∈ Pn | f (p) = 0, ∀f ∈ I}.
Observe that using Hilbert basis Theorem, we can consider a finite number of
homogeneous polynomials generating our ideal. So, we can understand the underlying topological set of this projective algebraic variety as the subset of Pn cut out
by the hyperplanes defined by the homogeneous polynomials generating our ideal.
However, observe that the variety contains much more information than that, for
example the polynomials ideals hxi and hx2 i define the same subset of P1 , which is
just the origin, but the second variety contain the origin with multiplicity two.
4
J. MORAGA
Example 2.1. (Hypersurfaces) Given any polynomial f in k[x0 , . . . , xn ], then if the
polynomial is irreducible, it defines a prime ideal hf i. The projective variety defined
by this ideal is called the hypersurface defined by f . For example, we can consider
the irreducible homogeneous polynomial x0 x3 − x1 x2 ∈ k[x0 , x1 , x2 , x3 ], which is a
quadratic polynomial in four variables, it will define the following hypersurface of
P3 :
X := {[z0 : z1 : z2 : z3 ] ∈ P3 | z0 z3 − z1 z2 = 0}.
We call such kind of hypersurfaces quadrics since they are defined by quadratic
polynomials. Analogously, we can define cubic and quadratic hypersurfaces. Now, if
we consider a new homogeneous polynomial, for example x0 , then we can construct
the variety corresponding to hx0 , x0 x3 − x1 x2 i, which will be the following:
Y := {[z0 : z1 : z2 : z3 ] ∈ P3 | z0 z3 − z1 z2 = 0, z0 = 0}.
Anyway, observe that the same ideal can be generated by x0 and x1 x2 , so the above
variety can be also described as
Y := {[z0 : z1 : z2 : z3 ] ∈ P3 | z1 z2 = 0, z0 = 0}.
Observe that Y is the union of two varieties, the first corresponding to hx1 , x0 i and
the second corresponding to hx2 , x0 i. Y is not irreducible since it can be written
as the effective union of projective varieties define by prime ideals (we say that
an union of sets ∪i Ci is effective is noone is containe in other). We will call Y a
projective variety as well, but we will call a variety irreducible if it is not the effective
union of varieties defined by prime ideals. Observe that, any variety defined by a
prime ideal is itself irreducible. Given a variety X that can be written as an effective
union of irreducible varieties, we will call such irreducible varieties the irreducible
components of X.
Moreover, observe that Y is a variety contained in X, and our intuition should
say to us that everytime that we impose a new equation to a variety, we are dropping
the dimension by one. Now, we will give formal definition of those concepts.
Definition 2.2. Given a projective variety X ⊂ Pn , and Y ⊂ X a subset, we will
say that Y is a subvariety of X if it is itself a variety.
Example 2.3. Observe that if we have the inclusion of a subvariety Y ⊂ X, being
Y and X irreducibles, then we have the opposite inclusion of the defining ideals.
Given an irreducible projective variety X any point [a0 : · · · : an ] = p ∈ X, it will
be always a subvariety of X, in fact, given I the ideal defining X, then we have
the following containment I ⊂ (x0 − a0 , . . . , xn − an ), being the second a maximal
homogeneous ideal of k[x0 , . . . , xn ], then we conclude the claim. In other words,
any variety defined by an ideal I will be the collection of points corresponding to
the maximal ideals that contain I in k[x0 , . . . , xn ].
Definition 2.4. Given a projective variety X, we define its dimension as the
supremum over all the integers numbers n ∈ Z≥0 , such that there exists a family
of proper contained subvarieties of X:
X = X0 ) X1 ) X1 ) · · · ) Xn .
Given a subvariety Y ⊂ X, we say that the codimension of Y inside X is equal to
dim(X) − dim(Y ). Observe that the points of Pn has maximal codimension, while
the hypersurfaces are subvarieties of codimension one.
NOTES ON COX RINGS
5
Remark 2.5. Observe that this definition can be given in terms of the defining
ideals as follows: A projective variety X ⊂ Pn , defined by a prime ideal I, has
dimension d, where d is the supremum between all the integers such that there
exists a family of proper contained ideals of k[x0 , . . . , xn ]:
I = I0 ( I1 ( I2 ( In .
Example 2.6. Any point of the projective space Pn has dimension zero with the
above definition, in fact, the point can not contain properly a subset. Moreover, a
hypersurface of Pn has dimension n − 1.
Definition 2.7. Given a projective variety X defined by the ideal I, we will call
the ring A(X) = k[x0 , . . . , xn ]/I, the ring of coordinates of X, or the ring of regular
functions of X.
Remark 2.8. The idea of the ring of regular functions of X is to recover all the
homogeneous polynomials of k[x0 , . . . , xn ] which can define subvarieties of Pn whose
restriction to X is non-trivial.
Definition 2.9. Given an irreducible projective variety X, we define the field of
rational functions as the zero-degree part of the field of fractions of A(X). In other
words, we define
f
| f, g ∈ A(X), deg(g) 6= 0, deg(f ) − deg(g) = 0 .
K(X) =
g
The idea of defining this field, follows the same idea given in remark 2.8.
Now, our purpose is to define the order of vanishing of a rational function of X
along a subvariety V ⊂ X. For this sake, we will need to introduce two things, first
we need the concept of generic points. Observe that Pn is the set of all maximal
homogeneous ideals k[x0 , . . . , xn ]. Now, we will consider a new space whose points
are all the homogeneous prime ideals of k[x0 , . . . , xn ], we will denote this space
as PnSch , called the Scheme of k[x0 , . . . , xn ], we will consider the topology on PnSch
whose closed subsets are of the form
V (I) = {J | J a prime homogeneous ideal with J ⊃ I},
where I is a prime homogeneous ideal. Observe that the underlying topological
sets of Pn and PnSch are very close. To obtain PnSch we have to add to Pn a point
ρV for each projective subvariety V ⊂ Pn , such that its closure {ρV } = V . Given
a primer homogeneous ideal I ∈ k[x0 , . . . , xn ] the set V (I) in PnSch is the projective
scheme associated to the projective variety V (I) ⊂ Pn . Consider a projective
variety X ⊂ Pn and a subvariety of codimension one V ⊂ X. Let V ⊂ PnSch be
the scheme associated to the subvariety V ⊂ Pn , we can consider the generic point
ρV ∈ PnSch , and define the local ring of ρV as follows
f
OρV =
| f, g ∈ k[x0 , . . . , xn ]k , k ∈ Z≥0 g 6∈ I ,
g
where k[x0 , . . . , xn ]k denotes the elements of degree k of this graded ring and I is
the ideal defining V . We define the order of vanishing of f along V as the length
of the OρV -module OρV /(f ).
Remark 2.10. In general, if we want to know the order of vanishing of a regular
function f in Pn along a point p, we don’t have to use the scheme-theoretic approach,
6
J. MORAGA
because p is itself a generic point of the subvariety {p}. In this setting, the vanishing
order of f at p is the maximal integer o number such that f ∈ mop , where mp ⊂
k[x0 , . . . , xn ] denotes the maximal ideal corresponding to the point p. In general,
a irreducible homogeneous polynomial f of degree d in k[x0 , . . . , xn ] has order of
vanishing d over its zero locus V (f )
Example 2.11. Consider the projective space P2 and two points p1 = [p11 : p21 : p31 ]
and p2 = [p12 : p22 : p32 ]. We want to count all the regular functions of degree 2 that
vanish with order at least 2 over p0 and at least 1 over p1 . First, after applying an
automorphism of P2 we can assume that p0 = [1 : 0 : 0] and p1 = [0 : 1 : 0]. We
know that the general form of a regular function over P2 of degree 3 is given by
f (x0 , x1 , x2 ) = c1 x20 + c2 x0 x1 + c3 x21 + c4 x2 x1 + c5 x2 x0 + c6 x22 ∈ k[x0 , x1 , x2 ].
When we assume that f vanishes with order at least one over [0 : 1 : 0] we have
that f (0, 1, 0) = 0, this is the same than saying that c3 = 0. Now, if we want f to
vanish with order at least two over [1 : 0] we have that
df
df
f (1, 0, 0) =
(1, 0, 0) =
(1, 0, 0) = 0,
dx2
dx1
which means that c1 = c2 = c5 = 0. Thus, the family of regular functions of P2
passing though two points with multiplicity 2 and 1 respectively is parametrized
by a family of polynomials
c4 x0 x2 + c6 x22 .
Observe that this family is two dimensional. In general, in Pn the dimension
of the
space parametrizing regular functions of degree d has dimension d+n
,
and
if we
n
ask this regular functions to vanish in the points p1 , . . . , pr with multiplicity
at
least
m1 , . . . , mr respectively, then the point number i is imposing mi +n−1
conditions
n
to the space regular functions. Anyway, this conditions can be linearly dependent,
so we have a lower bound for the dimension of the space of hypersurfaces of degree
d on Pn passing through points p1 , . . . , pr with multiplicity at least m1 , . . . , mr
respectively. The well-known expected dimension is the following
X
r n+d
mi + n − 1
−
.
n
n
i=1
Definition 2.12. The group of Weil
P divisors of a variety X is the abelian group
generated by all the formal sums i ni Di , where ni is an integral number and the
Di are irreducible subvarieties of codimension one in X. We denote the abelian
group of Weil divisors by WDiv(X).
Example 2.13. For example, consider the projective space Pn , then all the hypersurfaces, or subvarieties of codimension one, are defined by homogeneous polynomials in k[x0 , . . . , xn ], so the Weil group is a infinite dimensional torsion free abelian
group.
Definition 2.14. (Principal divisors) Given a rational function f on a projective
variety X, we will define the weil divisor associated to f as
X
(f ) =
ordD (f )D,
D
where the sum is taken over all the codimension one subvarieties of X, and ordD (f )
denotes the vanishing order of f at D. We call those divisors principal Weil divisors.
NOTES ON COX RINGS
7
We denote by PDiv(X) the abelian subgroup of WDiv(X) generated by all the Weil
divisors which are principal.
Example 2.15. Consider the rational function
f (x0 , x1 , x2 ) =
x0 x1 − x22
∈ K(P2 ).
x20 + x21
Since x0 x1 − x22 vanishes along the subvariety V1 = V (x0 x1 − x22 ) with order two
and x20 + x21 vanish with order one along the subvarieties V2 = V (x0 + ix1 ) and
V3 = V (x0 − ix1 ). Then we conclude that (f ) = V1 − V2 − V3 . Othe possible
example is
xn xm
2
g(x0 , x1 , x2 ) = 1n+m
∈ K(P2 ),
x0
whose corresponding Weil divisor is (g) = nV (x1 ) + mV (x2 ) − (n + m)V (x0 ).
Definition 2.16. (Divisor class group) Given a projective variety X we define the
Divisor class group of X as the quotient group WDiv(X)/ PDiv(X). We denote
such group by CDiv(X), in other words, two Weil divisors belongs to the same class
if and only if its difference is a principal divisor.
Example 2.17. Consider a morphism
φ : WDiv(Pn ) → Z defined in the following
Pn
way. Consider a Weil divisor i=1 ni Di , then every codimension one P
subvariety Di
n
n
of
P
is
defined
by
a
polynomial
of
degree
d
in
k[x
,
.
.
.
,
x
].
Then
φ
(
i
0
n
i=1 ni Di ) =
Pn
n
n
d
.
Observe
that
for
every
rational
function
f
∈
K(P
)
we
have that
i
i
i=1
Pn
φ((f )) = 0. On the other hand, given a Weil divisor D = i=1 ni Di such that
φ(D) = 0, and pick fi the polynomials defining Di , then we have that
D = (Πni=1 fini ) .
We conclude that ker(φ) = PDiv(Pn ), and then we have that CDiv(Pn ) ' Z.
Moreover, the class of an hyperplane H on Pn generates CDiv(Pn ). More general,
we have that CDiv(Pn1 ×· · ·×Pnk ) ' Zk , generated by the classes of the hyperplanes
in the i-th coordinates, for i ∈ {1, . . . , k}.
Given a divisor D ∈ WDiv(X) on a projective variety X, this class defines a
k-vector space given by
H 0 (X, D) = {f ∈ K(X) | (f ) + D ≥ 0},
where D ≥ 0 means that all the coefficients of the divisor D are non-negative, this
kind of divisors are called effective. Observe that given two Weil divisors D1 and D2
with the same class in the Class group, then we have that H 0 (X, D1 ) ' H 0 (X, D2 ).
This k-vector spaces are called Riemann-Roch spaces of the divisor D.
3. Cox Rings
Now, given a projective variety, we are interesed in calculate a basis for the
k-vector spaces H 0 (X, D) with D ∈ WDiv(X). Let’s start with some examples:
Example 3.1. Recall that CDiv(Pn ) is generated by the class of an hyperplane,
for example, fix the hyperplane x0 in k[x0 , . . . , xn ] and we can compute the corresponding Riemann-Roch spaces H 0 (Pn , dH) = k[x0 , . . . , xn ]d . Moreover, if we
denote by Hi the class of a hyperplane in the i-th coordinate of Pn1 × · · · × Pnk , we
have the following equality
H 0 (Pn1 × · · · × Pnk , d1 H1 + . . . dk Hk ) ' k[x10 , . . . , x1n1 , . . . , xk0 , . . . , xknk ](d1 ,...,dk ) ,
8
J. MORAGA
where the ring on the right represent the subring of k[x10 , . . . , x1n1 , . . . , xk0 , . . . , xknk ]
generated by monomials with degree di over the variables xi0 , . . . , xini . In particular,
observe that H 0 (Pn , dH) = 0 for d < 0 and the analogous statement holds for
product of projective spaces.
Definition 3.2. (Cox Rings) Let X be a projective variety whose Class group is
free and finitely generated, then the Cox ring of X is defined by
M
Cox(X) =
H 0 (X, [D]),
[D]∈WDiv(X)
where the multiplication by constants is defined by multiplicating on each RiemannRoch space and the multiplication of two elements f1 ∈ H 0 (X, [D1 ]) and f2 ∈
H 0 (X, [D2 ]) belongs to H 0 (X, [D1 + D2 ]). In other words, Cox(X) is naturally
graded by the Class group of X.
Remark 3.3. Cox rings can be constructed in a more general setting, but in the
more general cases the construction is non-trivial and requires some work to prove
the is well-defined. Most of the recent research in Cox Rings can be read in the
book [1].
Example 3.4. Using the above example we conclude that
Cox(Pn1 × · · · × Pnk ) = k[x10 , . . . , x1n1 , . . . , xk0 , . . . , xknk ].
In what follows, we will give some properties of the Cox rings, all the proofs can
be found in [1].
Theorem 3.5. There exists contravariant functors being essentially inverse to each
other between graded affine algebras and affine varieties with quasitorus action.
Under these equivalences the graded homomorphisms correspond to the equivariant
morphisms of varieties.
Remark 3.6. The above theorem says that Cox(X) has the action of a torus of
dimension rank(CDiv(X)). We will call such torus, the class group torus.
Definition 3.7. We define an ideal, called the irrelevant ideal of the Cox ring of
X:
M
Irr(X) =
H 0 (X, [D]).
[D]6=0
The following Theorem says that any projective variety can be recovered from
its Cox ring.
Theorem 3.8. Given a projective variety X with Cox ring Cox(X), then X is the
good quotient of Spec(Cox(X)) − V (Irr(X)) by the action of the class group torus.
The aim of the following example is to clarify the above theorems in a very
simple setting.
Example 3.9. Recall that Cox(Pn1 × · · · × Pnk ) = k[x10 , . . . , x1n1 , . . . , xk0 , . . . , xknk ].
Then, we have that
Spec(Cox(Pn1 × · · · × Pnk )) = k
Pk
i=1
ni
.
The class group torus is k-dimensional, and we will denote its coordinates by
(t1 , . . . , tk ). The action of the class group torus is given by multiplicating t1 to
NOTES ON COX RINGS
9
Pk
the first n1 variables of k i=1 ni , then t2 to the second n2 variables of k
so on. The irrelevant ideal is
+
*n
i
Y
i
xj | j ∈ {1, . . . , k} .
Pk
i=1
ni
, and
j=1
Observe that
Spec(Cox(Pn1 × · · · × Pnk )) − Irr(Cox(Pn1 × · · · × Pnk )) ' (k ∗ )n1 × · · · × (k ∗ )nk ,
and the quotient of this last set by the described action is just Pn1 × · · · × Pnk as
desired.
Remark 3.10. In general, every codimension one subvariety of X corresponds to
an element of the Cox ring and every rational function on X corresponds to the
quotient of two regular functions of the Cox ring having the same degree.
Example 3.11. Consider P1 × P1 , with its Cox ring k[x0 , x1 , y0 , y1 ]. Then we
can take two hypersurfaces of degree (2, 1) in P1 × P1 defined by the polynomials
x20 y0 − x21 y1 and x21 y0 − x20 y1 . Then the rational function
f (x0 , x1 , y0 , y1 ) =
x20 y0 − x21 y1
,
x21 y0 − x20 y1
is a well-defined function on P1 × P1 with coordinates ([x0 : x1 ], [y0 : y1 ]). This
kind of coordinates in P1 × P1 are called Cox coordinates.
Remark 3.12. For the construction of the blow-up of a projective space at a
point we will refer [2]. In general, we will recall that given a projective variety X
e → X such
and a point p ∈ X, there exists a projective birational morphism π X
that it is an isomorphism over X − {p} and the preimage of {p} is a projective
space of dimension dim(X) − 1, we call this preimage exceptional divisor over p.
e whose cokernel is
This morphism defines an injection π ∗ : CDiv(X) → CDiv(X)
e '
generated by the class of E. In particular, we have an isomorphism CDiv(X)
CDiv(X) ⊕ hEi. In general, we can blow-up r differents points p1 , . . . , pr in a
e with exceptional divisors
projective variety X and we will obtain a new variety X
E1 , . . . , Er and the following isomorphism holds
e = CDiv(X)
CDiv(X)
r
M
hEi i.
i=1
Then we have that
e =
Cox(X)
M
[D]∈CDiv(X),m1 ,...,mr ∈Zr
H
0
∗
π [D] −
r
X
!
mi Ei
.
i=1
e of degree ([D], m1 , . . . , mr ) corresponds to
The elements on the Cox ring Cox(X)
the elements of Cox(X) which vanish at the point pi with multiplicity mi .
Example 3.13. In this context, what we did in the example 2.11 was computing
e 2π ∗ H − 2E1 − E2 ), where π : X
e → P2
the dimension of the k-vector space H 0 (X,
2
is the blow-up of P at two points p1 and p2 , E1 and E2 are the corresponding
exceptional divisors and H is the class of an hyperplane in P2 . With the last
10
J. MORAGA
e → Pn the blow-up of Pn at
comment of example 2.11 we conclude that given π : X
r points, we have that
!! X
r
r X
n+d
mi + n − 1
0
∗
e dπ H −
dim H X,
m i Ei
≥
−
.
n
n
i=1
i=1
Example 3.14. (Returning to Nagata) We can identify the ring k[w1 , . . . , wr ] with
the Cox ring of Pr−1 , then we can consider the identification pj = [a1,j : · · · : ar,j ]
for 1 ≤ j ≤ n. We observe that a polynomial f ∈ k[w1 , . . . , wr ] is divisible by xm
i
if and only if it vanishes with order m at pi . In other words, the Nagata ring is
not finitely generated if and only if the Cox ring of the blow-up of Pr−1 at n points
is not finitely generated. This is known to hold for r = 3 and n ≥ 9 choosing the
points in very general position.
References
[1] Ivan Arzhantsev, Ulrich Derenthal, Jürgen Hausen, and Antonio Laface, Cox rings, Cambridge
Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015.
MR3307753 ↑8
[2] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate
Texts in Mathematics, No. 52. MR0463157 (57 #3116) ↑9
[3] David Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34, Springer-Verlag, Berlin-New York, 1965. MR0214602 (35 #5451)
↑2
Department of Mathematics, University of Utah, 155 S 1400 E, Salt Lake City, UT
84112
E-mail address: moraga@math.utah.edu