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Notes on Cox Rings

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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 2 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 2 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 : n2Z≥0 n2Z≥0 Since G is reductive, we can use Proposition 1.6 to write a decomposition Rn = G 0 Rn ⊕ Rn for each n 2 Z≥0, as vector spaces. Then, for each n we have a projection G morphism ρn : Rn ! Rn . The application ρ = ⊕n2Z≥0 is a well defined morphism ρ: R ! RG, with the following formal property (1.1) 8g 2 RG; f 2 R; ρ(gf) = gρ(f): G Consider the ideal I = ⊕n2Z≥0 Rn . Using Hilbert Basis Theorem we have that G there exists elements f1; : : : ; fk 2 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 2 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 2 R, given that h 2 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) 2 S. Thus, h 2 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 2 f1; : : : ; ng; yi 7! yi + tixi; i 2 f1; : : : ; ng; 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 2 f1; : : : ; ng we will use the following notation 0 1 X ai;jyj w = (x ····· x ) ; i 1 n @ x A j j for i 2 f1; : : : ; rg. 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) = fp 2 P j f(p) = 0; 8f 2 Ig: Observe that using Hilbert basis Theorem, we can consider a finite number of homogeneous polynomials generating our ideal. So, we can understand the under- lying 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 2 k[x0; x1; x2; x3], which is a quadratic polynomial in four variables, it will define the following hypersurface of P3: 3 X := f[z0 : z1 : z2 : z3] 2 P j z0z3 − z1z2 = 0g: 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 := f[z0 : z1 : z2 : z3] 2 P j z0z3 − z1z2 = 0; z0 = 0g: Anyway, observe that the same ideal can be generated by x0 and x1x2, so the above variety can be also described as 3 Y := f[z0 : z1 : z2 : z3] 2 P j z1z2 = 0; z0 = 0g: Observe that Y is the union of two varieties, the first corresponding to hx1; x0i and the second corresponding to hx2; x0i.
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