Notes on Cox Rings

NOTES ON COX RINGS

JOAQU´IN 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 Motivation1 2. Projective Varieties3 3. Cox Rings7 References 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 mathe- maticians 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 G the ring K = k(x1, . . . , xn) , 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 at- tempt to give a positive answer to Problem 1.2. Definition 1.4. Given a linear algebraic group G, we will call radical the compo- nent 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. More- over, 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 (kn, +) 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 G X G R = Rn,R = Rn . n∈Z≥0 n∈Z≥0

Since G is reductive, we can use Proposition 1.6 to write a decomposition Rn = G 0 Rn ⊕ Rn for each n ∈ Z≥0, as vector spaces. Then, for each n we have a projection G morphism ρn : Rn → Rn . 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). G Consider the ideal I = ⊕n∈Z≥0 Rn . Using Hilbert Basis Theorem we have that G there exists elements f1, . . . , fk ∈ R such that I = hf1, . . . , fki, 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 hifi, for hi ∈ R, given that h ∈ I. Using property 1.1, we conclude the following k ! k X X h = ρ(h) = ρ hifi = ρ(hi)fi. i=1 i=1

Since ρ(hi) are G-invariant elements of degree lower than h, by the induction hy- pothesis 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 (kn, +).

Example 1.8. (Nagata Counterexample) Consider the ring R = k[x1, . . . , xn, y1, . . . , yn] with the action of kn given by

xi 7→ xi, i ∈ {1, . . . , n},

yi 7→ yi + tixi, i ∈ {1, . . . , n}, n n−r where (t1, . . . , tn) denote the coordinates of k . Consider the subspace G = k ⊂ n P k , of codimension r, deﬁned by the equations j ai,jtj = 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 −1 the induced action of G in the ring R[x1 , . . . , xn ]. Observe the following equality −1 −1 −1 −1 y1 yn R[x1 , . . . , xn ] = k[x1 , . . . , xn ] ,..., . x1 xn The ring of invariants R[x−1, . . . , x−1]G is generated by the elements P ai,j yj and 1 n j xj moreover we have the following equality G −1 −1 G R = R ∩ R[x1 , . . . , xn ] . For each i ∈ {1, . . . , n} we will use the following notation   X ai,jyj w = (x ····· x ) , i 1 n  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 in the ring k[w1, . . . , wr] and m is a monomial in k[x1, . . . , xn] such that m is a 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 n X(I) = {p ∈ P | 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 hx2i 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 hfi. The projective variety defined by this ideal is called the hypersurface defined by f. For example, we can consider the irreducible homogeneous polynomial x0x3 − x1x2 ∈ k[x0, x1, x2, x3], which is a quadratic polynomial in four variables, it will define the following hypersurface of P3: 3 X := {[z0 : z1 : z2 : z3] ∈ P | z0z3 − z1z2 = 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, x0x3 − x1x2i, which will be the following: 3 Y := {[z0 : z1 : z2 : z3] ∈ P | z0z3 − z1z2 = 0, z0 = 0}.

Anyway, observe that the same ideal can be generated by x0 and x1x2, so the above variety can be also described as 3 Y := {[z0 : z1 : z2 : z3] ∈ P | z1z2 = 0, z0 = 0}.

Observe that Y is the union of two varieties, the first corresponding to hx1, x0i and the second corresponding to hx2, x0i. 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 ∪iCi 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 n homogeneous polynomials of k[x0, . . . , xn] which can define subvarieties of P 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 K(X) = | f, g ∈ (X), deg(g) 6= 0, deg(f) − deg(g) = 0 . g A 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 n n as PSch, called the Scheme of k[x0, . . . , xn], we will consider the topology on PSch 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 n n n n sets of P and PSch are very close. To obtain PSch we have to add to P a point n ρV for each projective subvariety V ⊂ P , such that its closure {ρV } = V . Given n a primer homogeneous ideal I ∈ k[x0, . . . , xn] the set V (I) in PSch is the projective scheme associated to the projective variety V (I) ⊂ Pn. Consider a projective n n variety X ⊂ P and a subvariety of codimension one V ⊂ X. Let V ⊂ PSch be the scheme associated to the subvariety V ⊂ Pn, we can consider the generic point n ρV ∈ PSch, and define the local ring of ρV as follows f O = | f, g ∈ k[x , . . . , x ] , k ∈ g 6∈ I , ρV g 0 n k Z≥0 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 o order of f at p is the maximal integer o number such that f ∈ mp, 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) 2 1 2 3 Example 2.11. Consider the projective space P and two points p1 = [p1 : p1 : p1] 1 2 3 and p2 = [p2 : p2 : p2]. 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 2 automorphism of P 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 2 2 2 f(x0, x1, x2) = c1x0 + c2x0x1 + c3x1 + c4x2x1 + c5x2x0 + c6x2 ∈ 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 2 which means that c1 = c2 = c5 = 0. Thus, the family of regular functions of P passing though two points with multiplicity 2 and 1 respectively is parametrized by a family of polynomials 2 c4x0x2 + c6x2. Observe that this family is two dimensional. In general, in Pn the dimension of the d+n space parametrizing regular functions of degree d has dimension n , and if we ask this regular functions to vanish in the points p1, . . . , pr with multiplicity at least mi+n−1 m1, . . . , mr respectively, then the point number i is imposing n conditions 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 n d on P passing through points p1, . . . , pr with multiplicity at least m1, . . . , mr respectively. The well-known expected dimension is the following r n + d X mi + n − 1 − . n n i=1 Definition 2.12. The group of Weil divisors of a variety X is the abelian group P generated by all the formal sums i niDi, 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 2 x0x1 − x2 2 f(x0, x1, x2) = 2 2 ∈ K(P ). x0 + x1 2 2 Since x0x1 − x2 vanishes along the subvariety V1 = V (x0x1 − x2) with order two 2 2 and x0 + x1 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 n m x1 x2 2 g(x0, x1, x2) = n+m ∈ K(P ), 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 niDi, then every codimension one subvariety Di n Pn of P is defined by a polynomial of degree di in k[x0, . . . , xn]. Then φ ( i=1 niDi) = Pn n i=1 nidi. Observe that for every rational function f ∈ K(P ) we have that Pn φ((f)) = 0. On the other hand, given a Weil divisor D = i=1 niDi such that φ(D) = 0, and pick fi the polynomials defining Di, then we have that

n ni D = (Πi=1fi ) . 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 H0(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 0 0 with the same class in the Class group, then we have that H (X,D1) ' H (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 H0(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 cor- 0 n responding Riemann-Roch spaces H (P , dH) = k[x0, . . . , xn]d. Moreover, if we n1 n denote by Hi the class of a hyperplane in the i-th coordinate of P × · · · × P k , we have the following equality H0( n1 × · · · × nk , d H + . . . d H ) ' k[x1, . . . , x1 , . . . , xk, . . . , xk ] , P P 1 1 k k 0 n1 0 nk (d1,...,dk) 8 J. MORAGA where the ring on the right represent the subring of k[x1, . . . , x1 , . . . , xk, . . . , xk ] 0 n1 0 nk generated by monomials with degree d over the variables xi , . . . , xi . In particular, i 0 ni observe that H0(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) = H0(X, [D]), [D]∈WDiv(X) where the multiplication by constants is defined by multiplicating on each Riemann- 0 Roch space and the multiplication of two elements f1 ∈ H (X, [D1]) and f2 ∈ 0 0 H (X, [D2]) belongs to H (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( n1 × · · · × nk ) = k[x1, . . . , x1 , . . . , xk, . . . , xk ]. P P 0 n1 0 nk 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) = H0(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( n1 × · · · × nk ) = k[x1, . . . , x1 , . . . , xk, . . . , xk ]. P P 0 n1 0 nk Then, we have that n n Pk n Spec(Cox(P 1 × · · · × P k )) = k i=1 i . 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 Pk ni ni the ﬁrst n1 variables of k i=1 , then t2 to the second n2 variables of k i=1 , and so on. The irrelevant ideal is

* ni + Y i xj | j ∈ {1, . . . , k} . j=1 Observe that n n n n ∗ n ∗ n Spec(Cox(P 1 × · · · × P k )) − Irr(Cox(P 1 × · · · × P k )) ' (k ) 1 × · · · × (k ) k , 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.

1 1 Example 3.11. Consider P × P , 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 2 2 2 2 x0y0 − x1y1 and x1y0 − x0y1. Then the rational function 2 2 x0y0 − x1y1 f(x0, x1, y0, y1) = 2 2 , x1y0 − x0y1 1 1 is a well-defined function on P × P 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 and a point p ∈ X, there exists a projective birational morphism πXe → X such 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. This morphism defines an injection π∗ : CDiv(X) → CDiv(Xe) whose cokernel is generated by the class of E. In particular, we have an isomorphism CDiv(Xe) ' CDiv(X) ⊕ hEi. In general, we can blow-up r differents points p1, . . . , pr in a projective variety X and we will obtain a new variety Xe with exceptional divisors E1,...,Er and the following isomorphism holds r M CDiv(Xe) = CDiv(X) hEii. i=1 Then we have that r ! M 0 ∗ X Cox(Xe) = H π [D] − miEi . r [D]∈CDiv(X),m1,...,mr ∈Z i=1

The elements on the Cox ring Cox(Xe) of degree ([D], m1, . . . , mr) corresponds to 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 0 ∗ 2 the dimension of the k-vector space H (X,e 2π H − 2E1 − E2), where π : Xe → P 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 comment of example 2.11 we conclude that given π : Xe → Pn the blow-up of Pn at r points, we have that r !! r 0 ∗ X n + d X mi + n − 1 dim H X,e dπ H − miEi ≥ − . n n i=1 i=1

Example 3.14. (Returning to Nagata) We can identify the ring k[w1, . . . , wr] with r−1 the Cox ring of P , then we can consider the identification pj = [a1,j : ··· : ar,j] m for 1 ≤ j ≤ n. We observe that a polynomial f ∈ k[w1, . . . , wr] is divisible by xi 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, J¨urgenHausen, 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 Grenzge- biete, 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: [email protected]