Actividades de promoción valorizadas y proyectadas en los años de la

Quotients, like those that define the supports, _Xr and X`, are difficult to deal with in an

algebraic geometry sense. So before we move on to deal with the conjecture proper, we will need to cover some basic results in geometric invariant theory. This is the language that is best for handling this problem.

Much of geometric invariant theory concerns itself with actions by reductive algebraic groups. It is fortunate thatHis a reductive group (sinceeS eis a reductive monoid).

Theorem 8.8. Let A be a finitely generated K-algebra. If G is a reductive group acting on A then AG is also a finitely generated K-algebra.

8.2. GeometricInvariantTheory 113

From geometric invariant theory we inherit two notions of quotient when a variety is acted upon by a reductive group. We will need to utilize both in order to tackle the question of projectivity of the left and right supports ofS.

Definition 8.9. Let G be a reductive group acting on variety X. Consider a variety, Y and an affine morphismφ : X → Y. If(Y, φ)satisfies the following properties it is called a good quotientof X by G.

(i)φis G-invariant (ii)φis surjective

(iii) if U ⊆ Y is open, thenφ∗:O(U)→ O(φ−1(U))is an isomorphism ofO(U)

ontoO(φ−1_(U))G

(iv) if W ⊆ X is closed and G-invariant thenφ(W)is closed

(v) if W1,W2 ⊆ X are disjoint, closed, and G-invariant, thenφ(W1)andφ(W2) are disjoint

If(Y, φ)is also an orbit space, it is called ageometric quotientof X by G.

Theorem 8.10. Let G be a reductive algebraic group and X an affine variety. Then there is an

affine variety, Y, and affine morphism,φ:X →Y, so that(Y, φ)is a good quotient of X by G. Proof. This is the content of the proof of Theorem 3.5 in [16].

Proposition 8.11.

(1) Let(Y, φ)be a good (geometric) quotient of projective variety, X, by reductive group, G. If U is open in Y, then(U, φ)is a good (geometric) quotient ofφ−1(U)by G. (2) Ifφ:X →Y is a morphism and{Ui}i∈Iis an open covering of Y such that(Ui, φ)is a

good (geometric) quotient ofφ−1_(U

i)by G for all i∈I, then(Y, φ)is a good (geometric) quotient of X by G.

Proof. This is just Proposition 3.10 in [16].

Proposition 8.12. Let(Y, φ) be a good quotient of X by G. If the action of G on X is closed, then(Y, φ)is a geometric quotient.

For this section of introductory material, we will limit our geometric invariant theory results to those that will explicitly apply to our proof of Renner’s conjecture. Although this will require some technical proofs that are basically reproductions of the work in [16], it saves us a discussion of linearisations and ample line bundles. To this end, consider the following scenario:

LetAbe an affine variety overKand letK∗_{act onA. Suppose thatAcontains acone point} a0. That is,a0 ∈ K∗yfor ally∈ A. On the level of our coordinate algebra, this turnsO(A) into

a nonnegatively graded algebra. A function f ∈ O(A) is calledhomogeneousof degree n, for somen∈_N, if for allk ∈K∗,y∈A, f(ky)= knf(y). If we denoteO(A)nto be the homogeneous

functions of degreen, thenO(A)=⊕n∈_NO(A)n.

We let P be the projective variety, (A\{a0})/K∗, with projection map, π : A\{a0} → P.

Suppose that we have a reductive group,G, that acts on Aby actionσwhich commutes with the action of K∗. Then the action ofGon Pis compatible with our projection map (it follows thatga0 = a0 for allg ∈ G). Our scenario can be summed up by the following commutative

diagram. G×A\{a0} A\{a0} G×P P id×π σ π σ

Our goal is to consider the supports as geometric quotients arising from exactly this situa- tion involvingAandP. So to get there, we now introduce the notions of stable and semi-stable elements inP.

Definition 8.13. For the situation of varieties A and P which we have defined, we say that a point x∈P is called,

(i)semi-stableif and only if there is a homogeneous function f ∈ O(A)Gof degree≥1such that f(x)_,0. By Psswe shall mean the set of all semi-stable elements in P.

(ii)stableif and only if x there is a homogeneous function f ∈ O(A)G of degree≥ 1such that f(x) _, 0and the action of G on Pf is closed (Pf = {x ∈ P | f(x) , 0}). By Ps we shall mean the set of all stable elements in P.

8.2. GeometricInvariantTheory 115

(iii)unstableif and only if it is not semi-stable.

Notice that the conditions of semi-stability and stability rely in part on homogeneous poly- nomials. So we can choose to show that for some ˆx∈Ain the fibre associated tox∈Psatisfies

f( ˆx)_, 0 when convenient. These sets of semi-stable and stable elements give us the following theorem, which establishes the existence of good and geometric quotients.

Theorem 8.14. There exists a good quotient,(Y, φ)of Pssby G, and Y is projective. Addition- ally, there exists an open subset, Ys _{⊆ Y such that} φ−1_(Ys₎ = _Ps _{and (Y}s, φ_{) is a geometric} quotient.

This proof is essentially a reproduction of Newstead’s proof of Theorem 3.14 (see [16]), but is specialized to our particular conditions.

Proof. Since we assumed the action ofGcommutes with the action ofK∗we can see that the action ofGonO(A), given byg·f(x)= f(g·x) preserves the degree of homogeneous functions. Indeed, (g· f)(k· x) = f(g·k· x) = f(k·g· x) = knf(g· x) = kn(g· f)(x). Thus,O(A)G is a homogeneous subalgebra ofO(A)=⊕n∈_NO(A)n.

For homogeneous f ∈ O(A)G _with

deg(f) ≥ 1, define Pf = {x ∈ P | f(x) , 0}. Notice

thatPss _:= S

f∈O(A)G_,deg(f)≥1P_f. EachP_f is an open affine subset ofP(being the compliment of the zero set of f). SinceG is reductive we know that there exists a good quotient ofPf byG.

(Theorem 8.10), (Yf, φf).

We can glue together all of these good quotients, (Yf, φf), to form a projective variety, Y,

and also get a map φ : S

f∈O(A)G_,deg(f)≥1Pf = Pss → Y. Our gluing maps should look like, hf f0 : φ_f(P_{f f}0) → φ_f0(P_{f f}0). As Brion notes in [5], this is achieved in a similar way that_Pn is

obtained from its open subsetsPnf. In this way we see thatφ|Pf=φf.

Observe that each (Yf, φf) is a good quotient ofPf = φ−1(Yf). We also note by our gluing

that {Yf}f∈O(A)G_,deg(f)≥1 is an open cover of Y (see 12.6 in [35]). Thus, by Proposition 8.11 it follows that (Y, φ) is a good quotient ofPss.

For the second part of our result, let Ys := φ(Ps) and let Y0 be the union of those Yf for

which the action of G on Pf is closed. It is clear that Ps ⊆ φ−1(Y0) and thus, Ys ⊆ Y0. By

tells us that (Y0, φ) is a geometric quotient ofP0 := φ−1(Y0). It follows thatPs = φ−1(Ys) and

Y0_\Ys= φ_(P0_\Ps_).

Thus,Y0_\Ys_{is closed inY}0_{by (iv) in our definition of a good quotient. SoY}s_{is open inY}0_.

Applying Proposition 8.11 again, we conclude that (Ys, φ) is a geometric quotient ofPs. Now that semistable and stable points give us quotients (as we will see, the quotients we will need to prove the conjecture) we devote the rest of this section to converting our definitions of semistable and stable into more useful forms. These forms are the ones used by C. S. Seshadri in [33] and are equivalent to Newstead’s and Mumford’s. First we need a lemma.

Lemma 8.15. Let G be a reductive group acting on affine variety, X. Let X1, X2be two disjoint, closed, G-invariant subsets of X. Then there is a function f ∈ O(X)G so that f(X1) = 0and

f(X2)= 1.

Proof. This is Lemma 3.3 in [16].

Here is our new definition for semistability.

Proposition 8.16. Let x∈P be an arbitrary element. Then x∈Pssif and only if a0 <Gx (hereˆ

ˆ

x represents a preimage of x underπ).

Proof. Let f ∈ O(A) be aG-invariant homogeneous function of degree≥1 such that f(x)_, 0. Observe that sincedeg(f)≥1, then f(ky) =kdeg(f)_{f(y) for ally∈Aandk∈ K}∗

, and by taking the closure, we can conclude that 0= f(a0). Then it is clear that f( ˆx), 0, and byG-invariance

f(y) is equal to a non-zero constant for ally∈Gxˆ. Hencea0 <Gxˆ.

Conversely, ifa0 <Gxˆ, then there exists by Lemma 8.15 aG-invariant function, f, such that f(a0)=0, f

Gxˆ=1. Then f has a constant term of 0, and it follows that some homogeneous part of f of degree≥1 must be nonzero at ˆx, soxis semi-stable.

In the same vein, we can find an equivalent condition for stability.

Lemma 8.17. An element x ∈ P is stable if and only if there exists a homogeneous function f ∈ O(A)G _{that has degree≥ 1such that f(x)}

, 0 and the morphismτf : G → Pf given by τf(g)=gx is proper.

8.2. GeometricInvariantTheory 117

Proof. This is Remark 3.16 (and Lemma 3.15) of [16].

Lemma 8.18. Let G be a linear group acting on variety, X. Then for x ∈ X, the morphism τ:G→ X, given byτ(g)=gx, is proper if and only if Gx is closed in X and Gxis finite.

Proof. This is Lemma 3.17 in [16].

Proposition 8.19. Let x∈ P be an arbitrary element. Then x∈ Ps _{if and only if|G} ˆ

x|< ∞and Gx is closed in A.ˆ

Proof. By Lemma 8.17 x ∈ P is stable if and only if there exists a homogeneous function

f ∈ O(A)G _{of degree ≥ 1 such that f(x)}

, 0 and the morphism τf : G → Pf given by τf(g)=gxis proper.

Fix an element, ˆx∈ Aover x. Letc = f( ˆx)_, 0, and defineC = {y∈ A| f(y)= c}, clearly a closed subvariety of A. Consider the morphismτf : G → C given by τf(g) = gx. It can be

seen thatτf =π◦τf. Lettingibe the inclusion map ofCintoAwe get, τf is proper ⇐⇒ π◦τf is proper (by equality)

⇐⇒ τf is proper (⇐follows fromπbeing proper)

⇐⇒ i◦τf :G→ Ais proper (sinceCis closed inA)

⇐⇒ τis proper (by equality)

So it follows that x∈ Pis stable if and only if there is a homogeneous function f ∈ O(A)G

that has degree≥ 1 such that f(x) _, 0 and the morphism τ : G → Agiven by τ(g) = gxis proper. By Proposition 8.16 this is equivalent toa0 <Gxˆandτbeing proper. Then, thanks to

Lemma 8.18, we can see that this is equivalent to the conditions thata0<Gxˆ,|Gxˆ|< ∞andGxˆ

is closed inA.

Now, ifGxˆis closed inA, it follows thata0<Gxˆ =Gxˆ, sinceG{a0}={a0}. So thenτbeing

proper impliesa0 <Gxˆ. Thus,x∈Pis stable if and only if|Gxˆ|<∞andGxˆis closed inA.

Geometric invariant theory will come in to play in our proof as our goal is to show that our right are left supports can be represented as Ps/G, and thus is the orbit space we want, but is also projective.