NOTES ON COX RINGS JOAQUÍN MORAGA

Transcription

1 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 1 2. Projective Varieties 3 3. Cox Rings 7 References 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(x 1,..., x n ) be the field of rational functions on n variables over the field k and K k(x 1,..., x n ) a subfield. Is K k[x 1,..., x n ] 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 R G 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[x 1,..., x n ], we can consider the ring K = k(x 1,..., x n ) G, and then deduce the Problem 1.2 with the case R = k[x 1,..., x n ]. 1

2 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 I n 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 which is also fixed by G and W W = 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 R G is finitely generated over k as well. Proof. We will prove the case R = k[x 1,..., x n ]. Observe that we have a Z 0 - grading on both rings R and R G. We will denote R = R n, R G = Rn G. n Z 0 n Z 0 Since G is reductive, we can use Proposition 1.6 to write a decomposition R n = Rn G R n for each n Z 0, as vector spaces. Then, for each n we have a projection morphism ρ n : R n Rn G. The application ρ = n Z 0 is a well defined morphism ρ: R R G, with the following formal property (1.1) g R G, f R, ρ(gf) = gρ(f). Consider the ideal I = n Z 0 Rn G. Using Hilbert Basis Theorem we have that there exists elements f 1,..., f k R G such that I = f 1,..., f k, We will prove that S, the subring generated by f 1,..., f k is the ring of invariants of R via the action G. Clearly, the inclusion S R G holds. On the other hand, pick h R G. We can assume by induction that all the elements of R G which have lower degree than h are in S. Observe that we can write h = k h if i, for h i R, given that h I. Using property 1.1, we conclude the following ( k ) k h = ρ(h) = ρ h i f i = ρ(h i )f i. Since ρ(h i ) are G-invariant elements of degree lower than h, by the induction hypothesis we conclude that ρ(h i ) 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

3 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[x 1,..., x n, y 1,..., y n ] with the action of k n given by x i x i, y i y i + t i x i, i {1,..., n}, i {1,..., n}, where (t 1,..., t n ) denote the coordinates of k n. Consider the subspace G = k n r k n, of codimension r, defined by the equations j a i,jt j = 0, with 1 i r. Moreover, we assume that the matrix A = (a i,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 x 1,..., x n are invariants by the action of G in R, we can consider the induced action of G in the ring R[x 1 ]. Observe the following equality 1,..., x 1 n R[x 1 1,..., x 1 n ] = k[x 1 1,..., x 1 n ] [ y1 x 1,..., y n x n The ring of invariants R[x 1 1,..., x 1 n ] G is generated by the elements j moreover we have the following equality R G = R R[x 1 1,..., x 1 n ] G. For each i {1,..., n} we will use the following notation w i = (x 1 x n ) a i,j y j, x j j ]. a i,jy j x j for i {1,..., r}. Observe that the algebra k[w 1,..., w r ] is contained in the ring of invariants R G, since all its generators are invariant elements. Finally, the ring R G is generated by all the elements of the form f m, where f is a homogeneous polynomial in the ring k[w 1,..., w r ] and m is a monomial in k[x 1,..., x n ] such that f 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[x 1,..., x n ], in P n. Recall that the zero locus of such polynomials does not depend on the representative that we take of a point of P n. In other words, given a homogeneous prime ideal I k[x 0,..., x n ], we have an associated variety X(I) = {p P n 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 P n 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 x and x 2 define the same subset of P 1, which is just the origin, but the second variety contain the origin with multiplicity two. and

4 4 J. MORAGA Example 2.1. (Hypersurfaces) Given any polynomial f in k[x 0,..., x n ], then if the polynomial is irreducible, it defines a prime ideal f. The projective variety defined by this ideal is called the hypersurface defined by f. For example, we can consider the irreducible homogeneous polynomial x 0 x 3 x 1 x 2 k[x 0, x 1, x 2, x 3 ], which is a quadratic polynomial in four variables, it will define the following hypersurface of P 3 : X := {[z 0 : z 1 : z 2 : z 3 ] P 3 z 0 z 3 z 1 z 2 = 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 x 0, then we can construct the variety corresponding to x 0, x 0 x 3 x 1 x 2, which will be the following: Y := {[z 0 : z 1 : z 2 : z 3 ] P 3 z 0 z 3 z 1 z 2 = 0, z 0 = 0}. Anyway, observe that the same ideal can be generated by x 0 and x 1 x 2, so the above variety can be also described as Y := {[z 0 : z 1 : z 2 : z 3 ] P 3 z 1 z 2 = 0, z 0 = 0}. Observe that Y is the union of two varieties, the first corresponding to x 1, x 0 and the second corresponding to x 2, x 0. 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 C i 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 P n, 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 [a 0 : : a n ] = 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 (x 0 a 0,..., x n a n ), being the second a maximal homogeneous ideal of k[x 0,..., x n ], 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[x 0,..., x n ]. 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 = X 0 X 1 X 1 X n. 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 P n has maximal codimension, while the hypersurfaces are subvarieties of codimension one.

5 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 P n, 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[x 0,..., x n ]: I = I 0 I 1 I 2 I n. Example 2.6. Any point of the projective space P n has dimension zero with the above definition, in fact, the point can not contain properly a subset. Moreover, a hypersurface of P n 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[x 0,..., x n ]/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[x 0,..., x n ] which can define subvarieties of P n 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 K(X) = { f g } f, g A(X), deg(g) 0, deg(f) deg(g) = 0. 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 P n is the set of all maximal homogeneous ideals k[x 0,..., x n ]. Now, we will consider a new space whose points are all the homogeneous prime ideals of k[x 0,..., x n ], we will denote this space as P n Sch, called the Scheme of k[x 0,..., x n ], we will consider the topology on P n Sch 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 P n and P n Sch are very close. To obtain Pn Sch we have to add to Pn a point ρ V for each projective subvariety V P n, such that its closure {ρ V } = V. Given a primer homogeneous ideal I k[x 0,..., x n ] the set V (I) in P n Sch is the projective scheme associated to the projective variety V (I) P n. Consider a projective variety X P n and a subvariety of codimension one V X. Let V P n Sch be the scheme associated to the subvariety V P n, we can consider the generic point ρ V P n Sch, and define the local ring of ρ V as follows { } f O ρv = g f, g k[x 0,..., x n ] k, k Z 0 g I, where k[x 0,..., x n ] 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 O ρv -module O ρv /(f). Remark In general, if we want to know the order of vanishing of a regular function f in P n along a point p, we don t have to use the scheme-theoretic approach,

6 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 m o p, where m p k[x 0,..., x n ] denotes the maximal ideal corresponding to the point p. In general, a irreducible homogeneous polynomial f of degree d in k[x 0,..., x n ] has order of vanishing d over its zero locus V (f) Example Consider the projective space P 2 and two points p 1 = [p 1 1 : p 2 1 : p 3 1] and p 2 = [p 1 2 : p 2 2 : p 3 2]. We want to count all the regular functions of degree 2 that vanish with order at least 2 over p 0 and at least 1 over p 1. First, after applying an automorphism of P 2 we can assume that p 0 = [1 : 0 : 0] and p 1 = [0 : 1 : 0]. We know that the general form of a regular function over P 2 of degree 3 is given by f(x 0, x 1, x 2 ) = c 1 x c 2 x 0 x 1 + c 3 x c 4 x 2 x 1 + c 5 x 2 x 0 + c 6 x 2 2 k[x 0, x 1, x 2 ]. 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 c 3 = 0. Now, if we want f to vanish with order at least two over [1 : 0] we have that f(1, 0, 0) = df dx 2 (1, 0, 0) = df dx 1 (1, 0, 0) = 0, which means that c 1 = c 2 = c 5 = 0. Thus, the family of regular functions of P 2 passing though two points with multiplicity 2 and 1 respectively is parametrized by a family of polynomials c 4 x 0 x 2 + c 6 x 2 2. Observe that this family is two dimensional. In general, in P n the dimension of the space parametrizing regular functions of degree d has dimension ( ) d+n n, and if we ask this regular functions to vanish in the points p 1,..., p r with multiplicity at least m 1,..., m r respectively, then the point number i is imposing ( m i+n 1) 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 d on P n passing through points p 1,..., p r with multiplicity at least m 1,..., m r respectively. The well-known expected dimension is the following ( ) n + d r ( ) mi + n 1. n n Definition The group of Weil divisors of a variety X is the abelian group generated by all the formal sums i n id i, where n i is an integral number and the D i are irreducible subvarieties of codimension one in X. We denote the abelian group of Weil divisors by WDiv(X). Example For example, consider the projective space P n, then all the hypersurfaces, or subvarieties of codimension one, are defined by homogeneous polynomials in k[x 0,..., x n ], so the Weil group is a infinite dimensional torsion free abelian group. Definition (Principal divisors) Given a rational function f on a projective variety X, we will define the weil divisor associated to f as (f) = ord D (f)d, D where the sum is taken over all the codimension one subvarieties of X, and ord D (f) denotes the vanishing order of f at D. We call those divisors principal Weil divisors.

7 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 Consider the rational function f(x 0, x 1, x 2 ) = x 0x 1 x 2 2 x x2 1 K(P 2 ). Since x 0 x 1 x 2 2 vanishes along the subvariety V 1 = V (x 0 x 1 x 2 2) with order two and x x 2 1 vanish with order one along the subvarieties V 2 = V (x 0 + ix 1 ) and V 3 = V (x 0 ix 1 ). Then we conclude that (f) = V 1 V 2 V 3. Othe possible example is g(x 0, x 1, x 2 ) = xn 1 x m 2 x n+m 0 K(P 2 ), whose corresponding Weil divisor is (g) = nv (x 1 ) + mv (x 2 ) (n + m)v (x 0 ). Definition (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 Consider a morphism φ: WDiv(P n ) Z defined in the following way. Consider a Weil divisor n n id i, then every codimension one subvariety D i of P n is defined by a polynomial of degree d i in k[x 0,..., x n ]. Then φ ( n n id i ) = n n id i. Observe that for every rational function f K(P n ) we have that φ((f)) = 0. On the other hand, given a Weil divisor D = n n id i such that φ(d) = 0, and pick f i the polynomials defining D i, then we have that D = (Π n f ni i ). We conclude that ker(φ) = PDiv(P n ), and then we have that CDiv(P n ) Z. Moreover, the class of an hyperplane H on P n generates CDiv(P n ). More general, we have that CDiv(P n1 P n k ) Z k, 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 D 1 and D 2 with the same class in the Class group, then we have that H 0 (X, D 1 ) H 0 (X, D 2 ). 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(P n ) is generated by the class of an hyperplane, for example, fix the hyperplane x 0 in k[x 0,..., x n ] and we can compute the corresponding Riemann-Roch spaces H 0 (P n, dh) = k[x 0,..., x n ] d. Moreover, if we denote by H i the class of a hyperplane in the i-th coordinate of P n1 P n k, we have the following equality H 0 (P n1 P n k, d 1 H d k H k ) k[x 1 0,..., x 1 n 1,..., x k 0,..., x k n k ] (d1,...,d k ),

8 8 J. MORAGA where the ring on the right represent the subring of k[x 1 0,..., x 1 n 1,..., x k 0,..., x k n k ] generated by monomials with degree d i over the variables x i 0,..., x i n i. In particular, observe that H 0 (P n, 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 Cox(X) = H 0 (X, [D]), [D] WDiv(X) where the multiplication by constants is defined by multiplicating on each Riemann- Roch space and the multiplication of two elements f 1 H 0 (X, [D 1 ]) and f 2 H 0 (X, [D 2 ]) belongs to H 0 (X, [D 1 + D 2 ]). 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(P n1 P n k ) = k[x 1 0,..., x 1 n 1,..., x k 0,..., x k n k ]. 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: Irr(X) = H 0 (X, [D]). [D] 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(P n1 P n k ) = k[x 1 0,..., x 1 n 1,..., x k 0,..., x k n k ]. Then, we have that Spec(Cox(P n1 P n k )) = k k ni. The class group torus is k-dimensional, and we will denote its coordinates by (t 1,..., t k ). The action of the class group torus is given by multiplicating t 1 to

9 NOTES ON COX RINGS 9 the first n 1 variables of k k ni, then t 2 to the second n 2 variables of k k ni, and so on. The irrelevant ideal is ni x i j j {1,..., k}. Observe that j=1 Spec(Cox(P n1 P n k )) Irr(Cox(P n1 P n k )) (k ) n1 (k ) n k, and the quotient of this last set by the described action is just P n1 P n k as desired. Remark 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 Consider P 1 P 1, with its Cox ring k[x 0, x 1, y 0, y 1 ]. Then we can take two hypersurfaces of degree (2, 1) in P 1 P 1 defined by the polynomials x 2 0y 0 x 2 1y 1 and x 2 1y 0 x 2 0y 1. Then the rational function f(x 0, x 1, y 0, y 1 ) = x2 0y 0 x 2 1y 1 x 2 1 y 0 x 2 0 y, 1 is a well-defined function on P 1 P 1 with coordinates ([x 0 : x 1 ], [y 0 : y 1 ]). This kind of coordinates in P 1 P 1 are called Cox coordinates. Remark 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 π X 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( X) whose cokernel is generated by the class of E. In particular, we have an isomorphism CDiv( X) CDiv(X) E. In general, we can blow-up r differents points p 1,..., p r in a projective variety X and we will obtain a new variety X with exceptional divisors E 1,..., E r and the following isomorphism holds CDiv( X) r = CDiv(X) E i. Then we have that Cox( X) = [D] CDiv(X),m 1,...,m r Z r H 0 ( π [D] ) r m i E i. The elements on the Cox ring Cox( X) of degree ([D], m 1,..., m r ) corresponds to the elements of Cox(X) which vanish at the point p i with multiplicity m i. Example In this context, what we did in the example 2.11 was computing the dimension of the k-vector space H 0 ( X, 2π H 2E 1 E 2 ), where π : X P 2 is the blow-up of P 2 at two points p 1 and p 2, E 1 and E 2 are the corresponding exceptional divisors and H is the class of an hyperplane in P 2. With the last

10 10 J. MORAGA comment of example 2.11 we conclude that given π : X P n the blow-up of P n at r points, we have that ( )) r ( ) n + d r ( ) dim (H 0 X, dπ mi + n 1 H m i E i. n n Example (Returning to Nagata) We can identify the ring k[w 1,..., w r ] with the Cox ring of P r 1, then we can consider the identification p j = [a 1,j : : a r,j ] for 1 j n. We observe that a polynomial f k[w 1,..., w r ] is divisible by x m i if and only if it vanishes with order m at p i. In other words, the Nagata ring is not finitely generated if and only if the Cox ring of the blow-up of P r 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, MR [2] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, Graduate Texts in Mathematics, No. 52. MR (57 #3116) 9 [3] David Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34, Springer-Verlag, Berlin-New York, MR (35 #5451) 2 Department of Mathematics, University of Utah, 155 S 1400 E, Salt Lake City, UT address: moraga@math.utah.edu