Quivers and moduli spaces of pointed curves of genus zero

We construct moduli spaces of representations of quivers over arbitrary schemes and show how moduli spaces of pointed curves of genus zero like the Grothendieck-Knudsen moduli spaces $\overline{M}_{0,n}$ and the Losev-Manin moduli spaces $\overline{L}_n$ can be interpreted as inverse limits of moduli spaces of representations of certain bipartite quivers. We also investigate the case of more general Hassett moduli spaces $\overline{M}_{0,a}$ of weighted pointed stable curves of genus zero.


Introduction
The main topic of this paper is the relation of moduli spaces of pointed curves of genus zero, in particular the Grothendieck-Knudsen moduli spaces M 0,n and the Losev-Manin moduli spaces L n but also more general Hassett moduli spaces of weighted pointed stable curves of genus zero M 0,a , to moduli spaces of representations of the following quivers Q n and P n with fixed dimension vector as indicated by the numbers at the vertices: Moduli spaces of representations M θ (Q) of a quiver Q with some fixed dimension vector are constructed via geometric invariant theory (GIT) and depend on the choice of a weight θ. The collection of M θ (Q) for θ representing the finitely many GIT equivalence classes forms an inverse system and as inverse limit we have a space lim ← −θ M θ (Q) independent of choice of weights.
In this paper we show over arbitrary base schemes (see theorems 3.14 and 3.15): Our methods also apply to more general Hassett moduli spaces M 0,a of weighted pointed stable curves of genus zero [Ha03]. In theorem 4.4 we show that any of these Hassett moduli spaces is a limit of GIT quotients over an area in the weight space (considering normalised weights): It is natural to work over Spec Z as the moduli spaces of pointed curves are defined over the integers, and so are moduli spaces of representations of quivers. However, we have to extend the construction of moduli spaces of representations of quivers in [Ki94] by constructing these moduli spaces over arbitrary base schemes (see theorem 1.8): Theorem. For a finite quiver Q with indivisible dimension vector and generic weight θ there is, over an arbitrary base scheme, a fine moduli space M θ (Q) of θ-stable representations.
To show the isomorphisms between moduli spaces of pointed curves of genus 0 and limits of GIT quotients we consider the GIT quotients for generic weights as fine moduli spaces, which is natural from the quiver perspective, and compare the functor of the limit to the moduli functor of pointed curves via the following steps: -The functor of the limit associates to a scheme Y the set of collections (s θ ) θ of θstable representations over Y for generic θ (i.e. corresponding to full-dimensional GIT equivalence classes) that satisfy certain relations coming from GIT classes of codimension one, see proposition 1.14 and (2.6) at the beginning of subsection 2.4. -In proposition 2.15 we express the relations between the representations (s θ ) θ in the limit functor by certain equations. Whereas the original relations are local in the weight space, we have equations relating pairs of representations for all generic θ. We can reduce these sets of equations by remark 2.17. -In the proofs of the result for the individual moduli spaces of pointed curves (theorems 3.14, 3.15 and 4.4) we restrict the collections (s θ ) θ to certain sets of generic weights. Considering representations as P 1 -bundles with n sections (see corollary 2.10 and 2.12), these weights θ i resp. θ T correspond to sets of three sections, such that a representation is stable with respect to this weight if and only if these three sections are disjoint.
-Normalising the representations with respect to the three sections for each of these weights, the functor of the inverse limit becomes a functor consisting of sections of P 1 that satisfy certain equations. We compare these functors of the limit with the functors of the moduli spaces of pointed curves which arise from a natural closed embedding of the moduli space into a product of P 1 and are described in theorems 3.9, 3.12 and 4.3.
There is the following known relation between moduli spaces of pointed curves of genus 0 and GIT quotients of (P 1 ) n . Working over the complex numbers, by [KSZ91] under some assumptions limits of GIT quotients of toric varieties (X, T ) by subtori of T are the same as Chow quotients. Toric Chow quotients can be described by fibre polytopes, and in the case of quotients of projective space by secondary polytopes [KSZ91], [GKZ]. The Losev-Manin moduli space L n is the projective toric variety corresponding to the (n − 1)dimensional permutohedron [LM00], which arises as secondary polytope of the product of simplices ∆ 1 × ∆ n−1 and as fibre polytope for the operation of G m on (P 1 ) n . By the Gelfand-MacPherson correspondence (see [Ka93a]) the GIT quotients (P 1 ) n by PGL(2) and the GIT quotients of the Grassmannian G(2, n) by the maximal torus (G m ) n ⊂ GL(n), and similarly the Chow quotients (P 1 ) n / / PGL(2) and G(2, n)/ /(G m ) n coincide over the complex numbers. By [Ka93a] the Chow quotients G(2, n)/ /(G m ) n coincide with the Hilbert quotients G(2, n)/ / /(G m ) n and by [Ka93b] M 0,n is the closure of the space of Veronese curves in the Hilbert scheme or Chow variety. This is used to show that the Chow quotient G(2, n)/ /(G m ) n is isomorphic to M 0,n , see [Ka93a,(4.1.8)]. Together the works [Ka93a], [Ka93b], [KSZ91] show that over the complex numbers M 0,n is the limit of GIT quotients (P 1 ) n / θ PGL(2).
Outline of the paper. In section 1 we construct fine moduli spaces of quiver representations over arbitrary schemes. We show in theorem 1.8 that for a finite quiver Q, an indivisible dimension vector d and a generic weight θ there is a fine moduli space M θ (Q, d), thus extending [Ki94]. To construct M θ (Q, d) we apply the theory of GIT (see [MFK]) over arbitrary base schemes, using the formulation in terms of stacks in [Al14]. The main point is to show that the natural group operation on the stable locus of the representation space is free and the morphism to the quotient a torsor, see proposition 1.7.
The first step to describe the functors of points of inverse limits of moduli spaces of quiver representations is taken in proposition 1.14 by describing the functor of the limit by the functors of the moduli spaces M θ (Q, d) for equivalence classes of generic weights θ subject to some relations.
Section 2 contains on the one hand some preparations and miscellaneous material concerning the quivers P n and Q n : structure of the weight spaces for the quivers P n and Q n (compare to [Ka93a, (1.2),(1.3)], [GKZ,7.3.C]); relation between Q n+2 and P n , their weights and stable representations (corresponds to the morphism M 0,n+2 → L n after choice of two of the n + 2 marked points); the relation of moduli spaces of representations for P n and Q n and GIT quotients of (P 1 ) n . The last two points, discussed in subsections 2.2 and 2.3, both derive from different presentations of certain stacks.
On the other hand subsection 2.4 is one of the central parts of this paper where in proposition 2.15 we describe the functor of the limit of moduli spaces of representations for the quivers P n and Q n in terms of families of representations (s θ ) θ satisfying certain equations, where the weights θ represent the equivalence classes of generic weights.
In section 3 we consider the Grothendieck-Knudsen moduli spaces and Losev-Manin moduli spaces. It is known that the moduli spaces L n and M 0,n can be embedded into a product of P 1 via the natural morphisms L n → L 2 ∼ = P 1 and M 0,n → M 0,4 ∼ = P 1 (see [BB11], [GHP88]). These closed embeddings have interpretations in terms of root systems of type A. In the case of the Losev-Manin moduli space L n ⊆ (P 1 ) ( n 2 ) each copy of P 1 corresponds to a root subsystem in A n−1 isomorphic to A 1 , whereas the equations come from the root subsystems isomorphic to A 2 , see [BB11]. In the case of the Grothendieck-Knudsen moduli space M 0,n ⊆ (P 1 ) ( n 4 ) each copy of P 1 corresponds to a root subsystem in A n−1 isomorphic to A 3 , whereas the equations come from the root subsystems isomorphic to A 4 . This way the moduli space M 0,n can also be interpreted as the cross-ratio variety of root systems of type A 3 in A n−1 , cf. [Sek94], [Sek96]. We summarise the results on these embeddings, its interpretation in terms of root systems and the related description of the functor of points of L n and M 0,n in theorems 3.9 and 3.12.
Proving that there are isomorphisms L n ∼ = lim ← −θ M θ (P n ) (theorem 3.14) and M 0,n ∼ = lim ← −θ M θ (Q n ) (theorem 3.15), we compare the moduli functors of pointed curves with the corresponding functors of the limit. We use the description of proposition 2.15 and simplify it, where the main step consists of restricting the family of representations (s θ ) θ to certain sets of weights of the form θ i resp. θ T .
In section 4 we apply the same methods to the Hassett moduli spaces to show that M 0,a ∼ = lim ← −θ<a M θ (Q n ) (theorem 4.4). Here some additions are necessary to take into account the notion of a-stability of n-pointed curves. On the quiver side this corresponds to restriction to weights in the convex polytope P (a) = {θ ∈ ∆(2, n) | θ < a}. We also need a description of the moduli functor M 0,a analogue to that of L n and M 0,n arising from an embedding into a product of P 1 , see theorem 4.3.
Notations and conventions. We work over a base scheme S, usually we think of S as Spec Z. We use the theory of schemes as developed in [EGA], [EGA1]. We sometimes write y ∈ Y for a geometric point y : Spec k → Y . We use the standard theory of algebraic stacks, see [SP]. We work with algebraic stacks in the fppf topology. Concerning stability of quiver representations we use the sign convention opposite to [Ki94].
1 Moduli spaces of quiver representations 1.1 Moduli of quiver representations over arbitrary schemes Let Q be a finite connected quiver with set of vertices Q 0 and set of arrows Q 1 . For an arrow α ∈ Q 1 we denote s(α) and t(α) the source and target, i.e. s(α) α −→ t(α).
We have the usual notions of homomorphisms of representations, subrepresentations etc.
Definition 1.2. The representation space of Q for the dimension vector d = (d q ) q∈Q 0 over the base scheme S is the affine space over S ) is the Hom-scheme that represents the contravariant functor ). Equivalently, the scheme R(Q, d) can be defined as the scheme that represents the contravariant functor on S-schemes Y → free representations with fixed basis of (Q, d) over Y .
On the representation space R(Q, d) the group scheme over S By [Ki94] the following definition of (semi)stability is equivalent to (semi)stability in the sense of GIT of [MFK] (with q det(g q ) −θq the character corresponding to θ). Definition 1.3. For a representation V of Q of dimension vector (d q ) q∈Q 0 over an algebraically closed field and a weight θ = (θ q ) q∈Q 0 ∈ Q Q 0 we define: (a) The representation V is θ-semistable if and only if q∈Q 0 θ q d q = 0 and every subrep- The functors on S-schemes In order to construct moduli spaces of representations, following [Ki94] we consider GIT quotients by the (geometrically) reductive group scheme GL(d) or G(d) (see [Al14, Section 9] for some results on (geometrically) reductive group schemes). GIT over arbitrary locally noetherian schemes has been worked out first in [Ses77], here we use the formulation of [Al14] in terms of adequate moduli spaces of Artin stacks.
Let M θ (Q, d) = R(Q, d)/ / θ G(d) the GIT quotient with respect to L = O R(Q,d) with the linearisation determined by the character θ, that is where p : R(Q, d) → S is the structure morphism. We have the quotient morphism is the quotient stack. The next theorem is a special case of [Al14].
(3) For an algebraically closed field k the morphism q induces a bijection (4) Assume that S is locally noetherian. Then M θ (Q, d) is of finite type over S and q * F is coherent for every coherent O X θ (Q,d) -module F .
(5) q is universal for morphisms to schemes, i.e. for an S-scheme Z the map Restricting to the open subscheme of stable points R θ-s (Q, d) ⊆ R θ (Q, d) there is the following result corresponding to the notion of geometric quotient.
is an adequate moduli space and in addition for algebraically closed fields k the map [X θ-s (Q, d)(k)] → M θ-s (Q, d)(k) is a bijection, where [X θ-s (Q, d)(k)] denotes the set of isomorphism classes of objects in X θ-s (Q, d)(k).
In addition to these standard GIT results, in our situation we have: Proof. We first show that the operation of G(d) has trivial stabilisers. For an S-scheme Since v is stable, all geometric fibres of St(v) = Aut Y (V ) are of dimension 1 (containing the diagonal group ∆), therefore all geometric fibres of End Y (V ) are 1-dimensional affine spaces, so End Y (V ) is an affine line bundle over Y . It follows that We show that the operation is proper, that is Ψ : ) is a proper morphism, using the valuative criterion [EGA, II, (7.3.8)]. We can assume we have the base scheme S = Spec Z. Let A be a discrete valuation ring with maximal ideal m, field of fractions K and k = A/m, and let t be a generator of The element g ∈ G(d)(K) corresponds to a family of matrices with coefficients in K up to a common factor in K * , so we can represent it by a family of matrices (M q ) q with coefficients in A and choose bases of V q and V ′ q such that all M q are diagonal with diagonal entries of the form t e . Let l be the minimal integer such that t l occurs as diagonal entry of some M q . Considering the matrices for the homomorphisms ν α , ν ′ α with respect to the chosen bases, the (i, j)-entries for the two matrices for a given α differ by a factor t e = 1 if the diagonal entries for the corresponding pairs of basis elements differ by such a factor. In this case modulo m one of the two (i, j)-entries vanishes. Let W ′ q ⊆ V ′ q be the submodules generated by those basis elements which correspond to the diagonal entries of the form t l , and similary (U q ) q the submodules of (V q ) q corresponding to the basis elements corresponding to the remaining diagonal entries. Then central fibres of the modules W ′ q ⊆ V ′ q form a subrepresentation W ′ k ⊆ V ′ k and the central fibres of the modules U q ⊆ V q form a subrepresentation U k ⊂ V k . If these were proper nonzero subrepresentations, then, because V k , V ′ k are θ-stable, both subrepresentations would satisfy θ(W ′ k ) < 0, θ(U k ) < 0, which is impossible. Therefore W ′ k = V ′ k , U k = 0. It follows that the element g ∈ G(d)(K) can be represented by matrices with coefficients in A \ m. These matrices are invertible as matrices over A and form the required element in G(d)(A).
The tautological G(d)-equivariant representation U R θ-s (Q,d) on R θ-s (Q, d) can be made into a G(d)-equivariant representation if and only if the dimension vector d is indivisible, i.e. the set of integers d q is coprime, see [Ki94]. In this case by descent along the G(d)- Proof. We denote the above functor F θ-s (Q, d) and q : is defined by V and a choice of basis of V , using that R θ-s (Q, d) with U R θ-s (Q,d) represents the functor of θ-stable free representations with fixed bases.
To show that the composition • p Y up to an element of G(d) that arises from comparision of the two bases coming from U R θ-s (Q,d) of these isomorphic representations. Composing with q, it follows f The equivalence relation on isomorphism classes of geometric points of X θ (Q, d) in (3) after theorem 1.5, or equivalently expressed as GIT-equivalence on R θ (Q, d)(k) defined by [Ki94] equivalent to S-equivalence of semistable representations as in the following definition.
Definition 1.9. Two representations V, V ′ of Q over an algebraically closed field are called S-equivalent if there are filtrations in the category of θ-semistable representations In particular, θ-stable representations are S-equivalent if and only if they are isomorphic.
Corollary 1.10. Via the morphism R θ (Q, d) → M θ (Q, d) any θ-semistable representation of (Q, d) over an S-scheme Y determines a morphism Y → M θ (Q, d) over S. This way for a geometric point s : Spec k → S the geometric points M θ (Q, d) s (k) over s are in bijection with the S-equivalence classes of θ-semistable representations of (Q, d) over k.

Decomposition of the weight space and limits of moduli spaces of quiver representations
In the case of GIT quotients of projective varieties X over an algebraically closed field the chamber structure of the space NS G (X) Q , where NS G (X) is the Néron-Severi group of G-equivariant line bundles, has been described in [Th96], [DH98], [Re00], and in the case X = R(Q, d) of quivers (without oriented cycles) in [HP02], [Ch08].
Let Q be a finite connected quiver, d = (d q ) q∈Q 0 a dimension vector, S be a scheme and R(Q, d) the representation space over S. We have the space Q Q 0 of fractional linearisations of the line bundle O R(Q,d) , the weight space The decomposition of H(Q, d) as described below is a finite decomposition by walls which are part of the hyperplanes H d ′ = { q θ q d ′ q = 0} for the finitely many dimension vectors Remark 1.11. Assume we have the base scheme S = Spec k, k an algebraically closed field. Assume that there is a θ ∈ H(Q, d) such that θ-stable representations exist. We call dimension vectors 0 < d ′ < d such that all representations of dimension vector d which are θ-stable with respect to some θ ∈ H(Q, d) have a subrepresentation of dimension vector d ′ a generic dimension vector. The existence of a subrepresentation of dimension vector d ′ is a closed condition on θ R θ-s (Q, d) (the operation of G(d) on R θ-s (Q, d) is proper by proposition 1.7). The intersection of the open half-spaces {θ | q θ q d ′ q > 0} ⊂ H(Q, d) for generic dimension vectors d ′ is the set of weights θ, such that θ-stable representations exist. Its closure, the intersection of the closed half-spaces {θ | q θ q d ′ q ≥ 0} for generic d ′ , is the full-dimensional convex polyhedral cone C(Q, d). Subquivers closed under successors give rise to generic dimension vectors. If Q has no oriented cycles, the cone C(Q, d) is strongly convex.
We denote the GIT equivalence class of θ by For the following standard results cf. [Re00], [Ch08].
Proposition 1.13. Assume S = Spec k, k an algebraically closed field. For θ ∈ C(Q, d) the set C θ is a convex polyhedral cone and relint(C θ ) = C θ . The cones C θ for θ ∈ C(Q, d) form a finite fan covering C(Q, d).
Over S the decomposition of H(Q, d) is the refinement of the decompositions of the fibres. Assume that over all geometric points of S stable representations for some θ exist.
We call the intersection of such a hyperplane with C(Q, d) a wall. Walls whose intersection with int C(Q, d) is nonempty are called inner walls, otherwise outer walls.
The GIT equivalence classes of generic weights are open sets called chambers. A weight θ is generic if and only if θ ∈ int C(Q, d) and θ is not contained in an inner wall.
Proposition 1.14. Let Q be a quiver and (d q ) q∈Q 0 be an indivisible dimension vector. Assume that over all geometric points of the base scheme S stable representations for some θ exist. Then the functor lim Proof. The limit functor is isomorphic to this functor since each τ ∈ C(Q, d) is contained in some C θ for a generic θ. This functor is a closed subfunctor of the product because M τ (Q, d) is a scheme separated over S (use [EGA1, (5.2.5)]).
2 Weight space decomposition and moduli spaces for some bipartite quivers 2.1 The quivers P n , Q n and the structure of the weight space We define the quivers P n and Q n with fixed dimension vectors (cf. introduction).
For these quivers with dimension vectors we determine the decomposition of the weight space as considered in subsection 1.2. In the case of P n and Q n the structure on the weight spaces does not vary over the geometric fibres of a given base scheme S.
In the following when studying the structure of GIT classes in the cone C(Q) ⊂ H(Q) for Q = Q n , P n we restrict to the intersection with a certain affine hyperplane. In the case Q n this yields the hypersimplex ∆(2, n), in the case P n the product of simplices ∆ 1 ×∆ n−1 .
Remark 2.2. Decomposition of the weight space for Q n . A weight for the quiver Q n is given by a tuple θ = (η, Omitting the coordinate η we have an isomorphism H(Q n ) ∼ = Q n , with basis e 1 , . . . , e n ∈ Q n and coordinates θ 1 , . . . , θ n . A semistable representation exists if and only if 0 ≤ θ i ≤ 1 2 n i=1 θ i for all i, this defines a full-dimensional strongly convex polyhedral cone C(Q n ) ⊂ H(Q n ). The affine hyperplane { n i=1 θ i = 2} intersects this cone in the hypersimplex For θ = (θ 1 , . . . , θ n ) ∈ ∆(2, n) a θ-semistable representation that is not θ-stable exists if and only if θ is contained in one of the following walls. The outer walls are of the form An inner wall W {J,J ∁ } intersects the boundary of ∆(2, n) as follows: do not intersect in the interior of ∆(2, n) if and only if the two partitions of {1, . . . , n} into two subsets are compatible in the sense that there is an inclusion between the sets J 1 , J ∁ 1 and J 2 , J ∁ 2 . In this case, if we We have the representation space Lemma 2.3. Let V = (s 1 , . . . , s n ) be a representation of Q n over an algebraically closed field that is semistable with respect to some weight in the interior of ∆(2, n), in particular all s i are nonzero. Let the partition {1, . . . , n} = l J l be defined by i, j ∈ J l for some l if and only if s i ∼ s j . The set Θ(V ) = {θ ∈ ∆(2, n) | V is θ-semistable} is a closed convex polytope in ∆(2, n) with the following properties: These walls do not intersect in the interior of ∆(2, n).
(ii) The sets of vertices and edges of Θ(V ) are subsets of the sets of vertices and edges of ∆(2, n).
is full dimensional if and only if there are ≥ 3 classes J l , or equivalently, V is stable with respect to some weight. Otherwise there are two classes J and J ∁ and Proof. Most of the lemma is obvious. For (ii) note that by remark 2.2 all intersections of inner and outer walls that bound Θ(V ) result in a wall of some hypersimplex.
The polytopes Θ(V ) are the realisable matroid polytopes in the sense of [Ka93a]. Θ(V ) corresponds to the stability set of V denoted Ω(V ) in [Re00].
Considering the quiver P n we have the representation space R(P n ) = (A 1 × A 1 ) n with operation of (G m × G m ) × (G m ) n . We write a representation of P n over a scheme Y as a tuple V = (s 1 , . . . , s n ) of sections of (A 1 × A 1 ) Y → Y where s i corresponds to (α i,1 , α i,2 ). We set s i ∼ s j if s i and s j are in the same G m -orbit and add the sections s 0 = (0, 1), s ∞ = (1, 0) which are are only used to compare with setions s i up to ∼.
Remark 2.4. Decomposition of the weight space for P n . A weight for the quiver P n is an element A semistable representation exists if and only if η 1 , η 2 ≤ 0 and θ 1 , . . . , θ n ≥ 0. This defines a full-dimensional cone in C(P n ) ⊂ H(P n ). The affine hyperplane defined by η 1 + η 2 = −1 intersects this cone in a product of simplices The outer walls are of the form Remark 2.5. The above notation implies that for τ ∈ relint(W J ) representations (s 1 , . . . , s n ) such that s j ∼ s ∞ = (1, 0) for j ∈ J and s i ∼ s 0 = (0, 1) for i ∈ J ∁ are strictly τ -semistable.
Lemma 2.6. Let V = (s 1 , . . . , s n ) be a representation of P n over an algebraically closed field that is semistable with respect to some weight in the interior of

Relation between representations of Q n+2 and P n
We show that the structure of GIT equivalence classes in ∆ 1 × ∆ n−1 ⊂ H(P n ) arises from the structure of GIT equivalence classes in ∆(2, n+2) in the neighbourhood of some vertex e a + e b and compare the corresponding stacks and moduli spaces of representations.
We observe that the projections of the walls of ∆(2, n + 2) near e a + e b coincide with the walls of ∆ 1 × ∆ n−1 ⊂ H(P n ). The walls meeting the interior of ∆(2, n + 2) a,b are the walls for schemes Y over a base scheme S, where s ′ i arises from s i by the base change that transforms s a to (1, 0) and s b to (0, 1), induce an isomorphism of stacks Proof. (a) The maps on the sets of morphisms are given by composition with the base change maps that transform s a , s b to (1, 0), (0, 1) and forgetting the automorphisms of the spaces corresponding to the vertices q a , q b . The inverse functor is given on the objects by maps (s i ) i =a,b → (s i ) i setting s a = (1, 0), s b = (0, 1) and on the morphisms by taking those automorphisms of the spaces corresponding to the vertices q a , q b such that s a , s b remain fixed. One checks that both compositions of these two functors are isomorphic to the identity functors.
All θ contained in this line segment are elements of the same GIT equivalence class exceptθ.
The isomorphism (2.2) identifies the structure sheaves of the two stacks. We have to verify that the additional multiplications by the corresponding characters coincide. It suffices to consider the groupoids of Y -valued points for S-schemes Y of the given groupoid schemes. The elements of (( and thus an isomorphism of their moduli spaces These isomorphisms determine an isomorphism of the inverse systems of the moduli spaces 2.3 Moduli spaces for P n , Q n and GIT quotients of products of P 1 We compare the moduli spaces of representations of Q n and P n to GIT quotients of (P 1 ) n by the quotient groups ( for all i has a PGL(2)-linearisation (O P 1 (−2) being the canonical sheaf of P 1 ), and if an element of Pic((P 1 ) n ) has a PGL(2)-linearisation then this linearisation is unique. Thus we can identify NS PGL(2) (( has a (unique) PGL(2)-linearisation and the equivariant line bundles O R * (Qn) with θlinearisation and O (P 1 ) n (θ 1 , . . . , θ n ) define isomorphic line bundles on the quotient stacks under the identification in (2.4).

2.4
The functor of inverse limits of quiver varieties for P n and Q n Let Q = Q n or Q = P n . Because all chambers are connected via the relative interiors of GIT equivalence classes of codimension 1, we can rewrite the contravariant functor on S-schemes (1.1) in proposition 1.14 as For a representation s of Q over an S-scheme Y stable with respect to some generic θ the polytopes Θ(s(y)) for the fibres s(y) over the geometric points y of Y defined in subsection 2.1 have the property that {y ′ | Θ(s(y)) ⊆ Θ(s(y ′ ))} are the points of an open subscheme of Y . Therefore for two representations s, s ′ we have an open subscheme If s θ , s θ ′ are part of a family over Y satisfying the conditions in (2.6), then by the conditions coming from the walls meeting the interior of the polytopes Θ(s θ (y)), Θ(s θ ′ (y)).
To study the conditions coming from walls that separate polytopes Θ(s(y)) we consider the schemes of representations of Q for weights in a GIT equivalence class C τ of codimension 1 and adjacent chambers C θ , C θ ′ . For Q = Q n C τ ∩ ∆(2, n) is contained in a wall of the form W {J,J ∁ } for some J ⊂ {1, . . . , n}, 2 ≤ |J| ≤ n − 2. For Q = P n C τ ∩ ∆ 1 × ∆ n−1 is contained in a wall of the form W = W J for some J ⊂ {1, . . . , n}, 1 ≤ |J| ≤ n − 1. In case Q = Q n let i ∈ J ∁ , j ∈ J, in case Q = P n let i = 0, j = ∞. We consider representations of Q over Y as tupels of sections (s 1 , . . . , s n ) of P 1 Y . Let Z τ ⊂ R τ (Q) be the closed subscheme of strictly τ -semistable points. Its geometric fibres over S consist of three orbits: the orbit such that both conditions s l = s j ⇔ l ∈ J and s k = s i ⇔ k ∈ J ∁ are satisfied, and the two orbits Z τ,J , Z τ,J ∁ where only the first resp. the second condition is satisfied. The representations corresponding to geometric points of Z τ are all S-equivalent. In case Q = Q n assume j∈J θ j > 1, then i∈J ∁ θ ′ i > 1 and Z τ, and are isomorphisms elsewhere. The case Q = P n is similar with projective spaces P contains Z τ . The algebra of invariant functions on U τ corresponds to the algebra of torus invariant functions in the O S -algebra generated bys l,1 s l,0 for l ∈ J ands k,0 s k,1 for k ∈ J ∁ . This O S -algebra of torus invariants is generated by f i,j k,l =s k,0sl,1 s k,1sl,0 for l ∈ J, k ∈ J ∁ . The function f i,j k,l can be written in terms of s as f i,j k,l = We express the condition in (2.6) for a family (s θ ) θ in terms of equations after fixing i, j such that s θ i , s θ j and s θ ′ i , s θ ′ j are disjoint in a neighbourhood of some y ∈ Y .
Let (s θ ) θ be a family of representations over Y as in (2.6). For s θ , s θ ′ we have the natural structure of a closed subscheme Z(s θ , s θ ′ ) ⊆ Y supported on the closed subset Y \U (s θ , s θ ′ ). Under the assumptions and notations of lemma 2.14 let y ∈ U i,j and assume k, l ∈ {1, . . . , n} such that s θ k = (0 : 1), (1 : 0), s θ ′ k = (0 : 1) and s θ ′ l = (0 : 1), (1 : 0), s θ l = (1 : 0). Because equations (2.7) hold in a neighbourhood of y the scheme defined by the equation s θ ′ k = (1 : 0) coincides with the scheme defined by the equation s θ l = (0 : 1) and does not depend on the choice of k, l. The scheme Z(s θ , s θ ′ ) can be defined in a neighbourhood of y by each of these equations.
The following proposition gives a geometric interpretation of the condition in (2.8) that two representations are related by equations (2.7), and indicates the relation between inverse limits of moduli spaces of representations for the quivers P n , Q n and moduli spaces of chains and trees of P 1 with marked points.
Proposition 2.16. Let s θ , s θ ′ be representations of Q = Q n or Q = P n over an S-scheme Y stable with respect to generic θ, θ ′ . Assume s θ , s θ ′ are related by equations (2.7). We consider s θ , s θ ′ as tupels of sections (s θ In case Q = P n let i = 0, j = ∞ and U i,j = Y . We choose homogeneous coordinates x θ 0 , x θ 1 and x θ ′ 0 , x θ ′ 1 of P 1 U i,j such that s θ i = (0 : 1), s θ j = (1 : 0) with respect to x θ 0 , x θ 1 and s θ ′ i = (0 : 1), s θ ′ j = (1 : 0) (a) Let C i,j ⊂ (P 1 ) 2 U i,j be the closed subscheme defined by the equations (2.9) for all k. For given k let U i,j,k is given by the single equation (2.9) for this k. (b) The curves C i,j ⊂ (P 1 ) 2 U i,j glue (with the appropriate coordinate changes) to a reduced curve C ⊂ (P 1 ) 2 Y flat over Y , which contains all pairs of sections (s θ l , s θ ′ l ), is isomorphic to P 1 U (s θ ,s θ ′ ) over U (s θ , s θ ′ ) ⊆ Y via its two projections, and degenerates exactly over the subscheme Z(s θ , s θ ′ ) ⊆ Y to a chain of two P 1 intersecting transversally such that each projection defines an isomorphism on one component and contracts the other to a reduced point. (c) The curve C induces morphisms P 1 Y ⇆ P 1 Y which degenerate to P 1 → pt. over Z(s θ , s θ ′ ) and which restrict to mutually inverse isomorphisms P 1 Proof. (a) Assume that s θ k (y) = (0 : 1), (1 : 0). We can further assume s θ ′ k (y) = (1 : 0) (similar: s θ ′ k (y) = (0 : 1)). Then in a neighbourhood of y for all l we have s θ x θ 0 x θ ′ 1 = s θ l,1 s θ ′ l,0 x θ 0 x θ ′ 1 using equation (2.9) for k and equation (2.7) for k, l. (b) To show that the curves C i,j glue, it suffices to show that the curves C i,j and C i,j ′ (and similar: C i,j and C i ′ ,j ) coincide over U i,j ∩ U i.j ′ after the base changes where a satisfies the relation s θ,i,j j ′ ,0 s θ,i,j ′ j,1 + as θ,i,j j ′ ,1 = 0 and s θ k = (s θ,i,j k,0 : s θ,i,j k,1 ) with respect to coordinates x θ,i,j 0 , x θ,i,j 1 such that (s θ,i,j i,0 : s θ,i,j i,1 ) = (0 : 1), (s θ,i,j j,0 : s θ,i,j j,1 ) = (1 : 0) (similar for θ ′ , j ′ ). Applying these base changes one verifies that the equations (2.9) hold for all k with respect to i, j if and only if they hold for all k with respect to i, j ′ . The properties of C follow from the properties of C i,j,k given by the single equation (2.9) for k, which are easy to verify. (c) follows from (b).
Remark 2.17. Proposition 2.16 allows to restrict the sets of equations required to hold in functors like (2.8) in proposition 2.15. All equations (2.7) relating s θ , s θ ′ hold if the equations with respect to choices of i, j hold such that the corresponding sets {y | s θ i (y) = s θ j (y), s θ ′ i (y) = s θ ′ j (y)} cover Y . Also, with respect to fixed i, j, all equations relating s θ , s θ ′ hold if those for certain l hold over {y | s θ l (y) = s θ i (y), s θ j (y)} or {y | s θ ′ l (y) = s θ ′ i (y), s θ ′ j (y)} and the corresponding sets cover Y . Y → stable n-pointed chains of P 1 over Y / ∼ where a stable n-pointed chain of P 1 over a scheme Y is a flat proper morphism C → Y with sections s 0 , s ∞ , s 1 , . . . , s n : Y → C such that the geometric fibres (C y , s 0 (y), s ∞ (y), s 1 (y), . . . , s n (y)) are stable n-pointed chains of P 1 over algebraically closed fields.
L n is toric, it compactifies the algebraic torus L n = (G m ) n / G m , the moduli space of n points in P 1 \ {0, ∞}. It has been shown in [LM00] that the moduli functor of stable n-pointed chains of P 1 is represented by a projective scheme of relative dimension n − 1, using an inductive construction of L n together with the universal curve which is isomorphic to L n+1 → L n . This morphism L n+1 → L n , studied in a similar setting in [Kn83], is a special case of the following morphisms that arise by forgetting sets of sections, see [BB11,Construction 3.15]. The Grothendieck-Knudsen moduli space M 0,n is the moduli space of stable n-pointed curves of genus 0, i.e. of stable n-pointed trees of P 1 . More generally, stable n-pointed curves of genus g occur already in [SGA7(1), I.5] and their moduli spaces and stacks have been systematically studied in [Kn83].
Definition 3.4. A stable n-pointed curve of genus 0 over an algebraically closed field is a tuple (C, s 1 , . . . , s n ) where C is a complete connected reduced curve C of genus 0 with at most ordinary double points, i.e. a tree of P 1 , s 1 , . . . , s n are closed points of C such that each s i is a regular point of C, s i = s j for i = j and each component of C has a least 3 special points (i.e. singular points and marked points s i ).
Definition -Theorem 3.5. ( [Kn83]). Let n ∈ Z ≥3 . The Grothendieck-Knudsen moduli space M 0,n is the fine moduli space of stable n-pointed curves of genus 0, i.e. M 0,n represents the moduli functor (denoted by the same symbol) Y → stable n-pointed curves of genus 0 over Y / ∼ where a stable n-pointed curve of genus 0 over a scheme Y is a flat proper morphism C → Y with sections s 1 , . . . , s n : Y → C such that the geometric fibres (C y , s 1 (y), . . . , s n (y)) are stable n-pointed curves of genus 0.
M 0,n compactifies the moduli space M 0,n of n distinct points in P 1 . The fact that the moduli functor of stable n-pointed curves of genus 0 is represented by a projective scheme of relative dimension n − 3 has been shown in [Kn83] using an inductive argument on n, showing that the universal family over M 0,n is formed by the morphism M 0,n+1 → M 0,n . This morphism is a special case of the following morphisms for inclusions I ⊂ {1, . . . , n} which can be defined by mapping a stable n-pointed tree (C → Y, s 1 , . . . , s n ) to its image under the morphism defined by the sheaf ω C/Y ( i∈I s i ), where ω C/Y is the dualising sheaf (see [Kn83]).

Embeddings into products of P 1 and relation to root systems of type A
We consider the cross-ratio varieties for root subsystems of type A 3 in A n−1 defined and studied in [Sek94], [Sek96]. It was already observed in [Sek96] that the cross-ratio variety for root subsystems of type A 3 in A n−1 should be isomorphic to the Grothendieck-Knudsen moduli space M 0,n .
The root lattice of A n is the sublattice M (A n−1 ) ⊂ L(A n−1 ) = Z n generated by the roots α i,j = e i − e j where i, j ∈ {1, . . . , n} and e 1 , . . . , e n are the standard basis of Z n . We have the lattice L(A n−1 ) * dual to L(A n−1 ) and the lattice N (A n−1 ) = L(A n−1 ) * /(1, . . . , 1)Z dual to M (A n−1 ).
We consider cross-ratio varieties for root systems of type A 3 in A n−1 . Let A(L(A n−1 )) be the n-dimensional affine space with coordinates t 1 , . . . , t n corresponding to the basis e 1 , . . . , e n . We have the open subscheme U (A n−1 ) = A(L(A n−1 )) \ i =j {t i = t j }. For root subsystems A 3 ∼ = ∆ ⊆ A n−1 consisting of roots α i,j for i, j ∈ V = {i 1 , . . . , i 4 } ⊆ {1, . . . , n}, |V | = 4 there is the morphism depending on the choice of an ordering V = {i 1 , i 2 , i 3 , i 4 } or equivalently a simple system α i 1 ,i 2 , α i 2 ,i 3 , α i 3 ,i 4 in ∆ ∼ = A 3 . Considering the case n = 4, the image of each of these morphisms is X A 3 ,A 3 = P 1 \ {(0 : 1), (1 : 0)} and we denote its closure X A 3 ,A 3 . Each of these morphisms c i 1 ,i 2 ,i 3 ,i 4 A 3 ,A 3 fixes homogeneous coordinates on X A 3 ,A 3 or equivalently defines an isomorphism ϕ i 1 ,i 2 ,i 3 ,i 4 A 3 : X A 3 ,A 3 → P 1 . In the general case for root subsystems such that for every choice of an ordering Definition 3.7. We denote the image X A n−1 ,A 3 = A 3 ∼ =∆⊆An−1 c A n−1 ,∆ U (A n−1 ) and define the cross-ratio variety X A n−1 ,A 3 for root systems of type A 3 in A n−1 as the closure Let c A n−1 ,∆ : X A n−1 ,A 3 → X ∆,∆ be the morphisms induced by the projection on the factors of the product.
In a similar way we can construct a scheme from the set of root subsystems of type A 1 in A n−1 . We have the n-dimensional projective space P(L(A n−1 )) with homogeneous coordinates t 1 , . . . , t n corresponding to the basis e 1 , . . . , e n . The open subscheme T (A n−1 ) = P(L(A n−1 )) \ i {t i = 0} ⊂ P(L(A n−1 )) is the algebraic torus with character lattice M (A n−1 ). For root subsystems A 1 ∼ = {±α i,j } ⊆ A n−1 we have morphisms depending on the choice of an ordering of {i, j} or equivalently on the choice of a simple system α i,j in {±α i,j }. Considering the case n = 2, the morphisms c i,j A 1 ,A 1 are open embeddings with image X A 1 ,A 1 = P 1 \ {(0 : 1), (1 : 0)} and we denote its closure X A 1 ,A 1 . Each of these morphisms c i,j A 1 ,A 1 fixes homogeneous coordinates on X A 1 ,A 1 or equivalently defines an isomorphism X A 1 ,A 1 ∼ = P 1 . In the general case for root subsystems The morphisms c A n−1 ,{±α i,j } are homomorphisms of algebraic tori. The homomorphism of tori A 1 ∼ ={±αi,j }⊆A n−1 c A n−1 ,{±α i,j } embeds T (A n−1 ) into the open dense torus of j }⊆A n−1 P 1 coincides with the toric variety X(A n−1 ) associated with the root system A n−1 , i.e. its fan is the fan of Weyl chambers of the root system A n−1 in the lattice N (A n−1 ), because on the open dense torus T (A n−1 ) ⊂ X(A n−1 ) the morphisms c A n−1 ,{±α i,j } coincide with the morphisms for inclusions of root subsystems {±α ij } ⊆ A n−1 of type A 1 and the morphism {±α ij } γ {±α ij } is a closed embedding (see [BB11] or theorem 3.9 below). Already in [LM00, (2.6.3)] the Losev-Manin moduli space L n has been identified as the smooth projective toric variety associated with the (n − 1)-dimensional permutohedron, which coincides with X(A n−1 ). In [BB11] we give an alternative proof that the functor L n is representable and the fine moduli space L n is isomorphic to the smooth projective toric variety X(A n−1 ) after a systematic study of toric varieties X(R) associated with root systems R, their functorial properties with respect to maps of root systems and their functor of points, and show that the morphisms γ R : X(A n−1 ) → X(R) ∼ = X(A k−1 ) corresponding to inclusions of root systems A k−1 ∼ = R ⊂ A n−1 can be identified with the contraction morphisms γ I : L n → L I ∼ = L k determined by subsets I ⊂ {1, . . . , n}, |I| = k.  1). The remaining section s i j gives a section (s i j,0 : s i j,1 ) of P 1 Y . This determines an isomorphism L 2 ∼ = P 1 . We have (s i j,0 : s i j,1 ) = (s j i,1 : s j i,0 ) and the two choices of homogeneous coordinates are related by (x i j,0 : x i j,1 ) = (x j i,1 : x j i,0 ). We can identify L 2 = X(A 1 ) by (x j i,0 : where the product is over all subsets I = {i, j} ⊂ {1, . . . , n}, |I| = 2 or equivalently over all root subsystems A 1 ∼ = {±α i,j } ⊆ A n−1 , is a closed embedding. Its image in {i,j} L {i,j} = {±α i,j } X({±α i,j }) is given by the equations of the form for all subsets J = {i, j, k} ⊆ {1, . . . , n}, |J| = 3 or equivalently all root subsystems A 2 ∼ = {±α i,j , ±α j,k , ±α i,k } ⊆ A n−1 , where x i j,0 , x i j,1 are the homogeneous coordinates of L {i,j} = X({±α i,j }) as defined in example 3.8. The functor of points of L n = X(A n−1 ) is isomorphic to the contravariant functor Similarly, we can describe M 0,n and its embedding into a product of P 1 in terms of the root system A n−1 . The space U (A n−1 ) = A(L(A n−1 )) \ i =j {t i = t j } parametrises n distinct points in A 1 = P 1 \ {(1 : 0)}. The tautological family determines a morphism ψ : U (A n−1 ) → M 0,n . The following lemma is easy to verify.
corresponding to generators of the permutation group. The sections in these coordinates are related by for permutations of the upper indices and by (3.4) The following theorem follows mostly from [GHP88]. We use this closed embedding of M 0,n to describe its functor of points, add the interpretation in terms of root systems and the link to the cross-ratio varieties. To relate M 0,n to X A n−1 ,A 3 use lemma 3.10.
Theorem 3.12. The morphism where the product is over all subsets I ⊆ {1, . . . , n}, |I| = 4, is a closed embedding. This embedding induces an isomorphism M 0,n → X A n−1 ,A 3 and can be identified with the closed embedding where the product is over all root subsystems ∆ isomorphic to A 3 . The image of M 0,n ∼ = X A n−1 ,A 3 in I M 0,I ∼ = ∆ X ∆,∆ is given by the equations of the form Remark 3.13. The stable n-pointed tree can be reconstructed from data as in (3.5) over an S-scheme Y as the curve C ⊆ ( T P 1 ) Y , where the product is over subsets T ⊆ {1, . . . , n} such that |T | = 3, defined by the equations are the homogeneous coordinates of the factor corresponding to T = {i 1 , i 2 , i 3 } and related under permutations of elements of T by equations as the first two in (3.2) leaving out the lower index, and the sections are given as . This follows easily from theorem 3.12 and the fact that M 0,n+1 → M 0,n forms the universal n-pointed tree over M 0,n . One may also compare this to proposition 2.16.

Losev-Manin and Grothendieck-Knudsen moduli spaces as limits of moduli spaces of quiver representations
We relate moduli spaces of representations of P n to the Losev-Manin moduli spaces L n .
Theorem 3.14. There is an isomorphism Proof. By proposition 2.15 lim ← −θ M θ (P n ) ∼ = Lim(P n ). Consider the functor Lim ′ (P n ) defined by We claim that the natural morphism of functors Lim(P n ) → Lim ′ (P n ) that arises by restricting elements and relations is an isomorphism. The property int Θ(s θ (y)) ∩ Θ(s θ ′ (y)) = ∅ ⇐⇒ s θ ∼ = s θ ′ in a neighbourhood of y of (s θ ) θ ∈ Lim(P n )(Y ) follows from the equations relating s θ , s θ ′ and thus also holds for elements of Lim ′ (P n )(Y ): by proposition 2.16 we have an isomorphism s θ ∼ = s θ ′ on an open set and for y in its complement there are J, J ′ such that (s θ j (y)) j∈J coincide, (s θ ′ j (y)) j∈J ′ coincide and (J ∪ J ′ ) ∁ = ∅, which implies that Θ(s θ (y)) and Θ(s θ ′ (y)) are separated by a wall. By this property for elements of Lim(P n )(Y ) resp. Lim ′ (P n )(Y ) the polytopes Θ(s θ (y)) resp. Θ(s θ i (y)) have either disjoint interiors or coincide. We construct the inverse morphism by showing that there is a unique way to extend an element (s θ i ) i ∈ Lim ′ (P n )(Y ) to an element (s θ ) θ ∈ Lim(P n )(Y ). Such a (s θ ) θ ∈ Lim(P n )(Y ) is uniquely determined by its restriction to Lim ′ (P n )(Y ) because for any generic θ we have i {y | s θ (y) θ i -stable} = Y . On the other hand we can extend a given (s θ i ) i ∈ Lim ′ (P n )(Y ) to an element of Lim(P n )(Y ) if for any generic θ we have i {y | s θ i (y) θ-stable} = Y or equivalently for all y ∈ Y the polytopes Θ(s θ i (y)) cover ∆ 1 ×∆ n−1 . The fibres (s θ i (y)) i over a geometric point y ∈ Y define a decomposition of ∆ 1 × ∆ n−1 into polytopes: the polytopes Θ(s θ i (y)) and possibly remaining parts. By remark 2.4 and lemma 2.6 each of the remaining parts is of the form P (J, i.e. bounded by walls W J and W (J ′ ) ∁ , for some J, J ′ such that J (J ′ ) ∁ . But any such P (J, J ′ ) contains θ i for i ∈ {1, . . . , n} \ (J ∪ J ′ ). So already the polytopes Θ(s θ i (y)) cover ∆ 1 × ∆ n−1 . Thus we have shown Lim(P n ) ∼ = Lim ′ (P n ). By remark 2.17 we can restrict the equations s θ i k,0 s θ i l,1 s θ j k,1 s θ j l,0 = s θ i k,1 s θ i l,0 s θ j k,0 s θ j l,1 in the functor Lim ′ (P n ) relating s θ i , s θ j to equations with l = i. Thus Lim ′ (P n ) is isomorphic to the functor Lim ′′ (P n ) defined by Each s θ i consists of a family (s θ i j ) j=1,...,n of sections s θ i j : Y → P 1 Y up to operation of G m on P 1 Y fixing (0 : 1), (1 : 0). Within its isomorphism class we can choose this family such that s θ i i = (1 : 1), then we denote This functor is the same as the functor (3.1) in theorem 3.9.
In the same way we study the relation of moduli spaces of representations of Q n to the Grothendieck-Knudsen moduli spaces M 0,n .
For T ⊆ {1, . . . , n}, |T | = 3 we define the weight θ T for the quiver Q n as θ T i = 2 3 (1 − ε) for i ∈ T and θ T i = 2 n−3 ε for i ∈ T , where 1 ≫ ε > 0. The weights θ T are generic and for any θ T -stable s θ T the sections (s θ T i ) i∈T are disjoint. The moduli spaces M θ T (Q n ) are isomorphic to (P 1 ) n−3 .
Theorem 3.15. There is an isomorphism Proof. By proposition 2.15 lim ← −θ M θ (Q n ) ∼ = Lim(Q n ). Consider the functor Lim ′ (Q n ) defined by In the same way as in the proof in the case of P n we show that the natural morphism Lim(Q n ) → Lim ′ (Q n ) is an isomorphism by showing that T {y | s θ T (y) θ-stable} = Y for all (s θ T ) T defining an element of Lim ′ (Q n )(Y ) and all generic θ. For such (s θ T ) T and fixed y ∈ Y again the polytopes Θ(s θ T (y)) either coincide or their interiors are disjoint, and ∆(2, n) decomposes into the polytopes Θ(s θ T (y)) and remaining parts. Using remark 2.2 and lemma 2.3 we see that the remaining parts consist of polytopes bounded by walls W {J l ,J ∁ l } which do not intersect in the interior of ∆(2, n) and thus are of the form l {θ ∈ ∆(2, n) | j∈J l θ j ≤ 1} for disjoint subsets J l ⊂ {1, . . . , n}, 2 ≤ |J l | ≤ n − 2. Any such subset which is full-dimensional contains a θ T , thus T Θ(s θ T (y)) = ∆(2, n) for fixed y ∈ Y , which is equivalent to the above statement.
In the functor Lim ′ (Q n ) the statement that equations 2.7 hold means that for all T, T ′ and for all i, j locally over y | s θ T i (y) = s θ T j (y), s θ T ′ i (y) = s θ T ′ j (y) ⊆ Y , after choice of coordinates such that s θ T i , s θ T ′ i = (0 : 1) and s θ T j , s θ T ′ j = (1 : 0), for all k, l the equations hold.
We claim that we can restrict this set of equations to the set of equations for T, T ′ with |T ∩ T ′ | = 2. In the last step we have shown that for all y ∈ Y the polytopes Θ(s θ T (y)) cover ∆(2, n). By the last part of the proof of lemma 2.14 if the union of the walls separating Θ(s θ T (y)), Θ(s θ T ′ (y)) and Θ(s θ T ′ (y)), Θ(s θ T ′′ (y)) are exactly the walls that separate Θ(s θ T (y)), Θ(s θ T ′′ (y)) and if s θ T , s θ T ′ and s θ T ′ , s θ T ′′ in a neighbourhood of y are related by equations of the form (3.6), then so are s θ T , s θ T ′′ . The case that Θ(s θ T (y)) and Θ(s θ T ′′ (y)) coincide is trivial. It follows that it is enough to consider the situation that Θ(s θ T (y)), Θ(s θ T ′′ (y)) are separated by a wall W {J,J ∁ } = Θ(s θ T (y)) ∩ Θ(s θ T ′′ (y)). If T ⊆ J then there isT such that θT∈ Θ(s θ T (y)), |T ∩T | = 2,T ∩ J ∁ = ∅, thus we can assume that both T and T ′′ have nonempty intersection with J and J ∁ . In this case if ). Having shown that in the functor Lim ′ (Q n ) the equations (3.6) with |T ∩ T ′ | = 2 are sufficient, by remark 2.17 we can restrict to the equations with choice of i, j such that Also by remark 2.17 it suffices to consider the equations with k = i 4 and write i 5 for l. There are two cases i 3 = i 5 and i 3 = i 5 giving rise to two sets of equations. This shows that Lim ′ (Q n ) is isomorphic to the functor Lim ′′ (Q n ) defined by where we choose coordinates such that s θ {i 1 ,i 2 ,i 3 } The functor that associates to an S-scheme Y families of sections s i 1 ,i 2 ,i 3 i 4 : Y → P 1 Y for i 1 , i 2 , i 3 , i 4 ∈ {1, . . . , n}, |{i 1 , i 2 , i 3 }| = 3 that satisfy these equations is functor (3.5).  = (a 1 , . . . , a n ) ∈ Q n satisfies 0 < a i ≤ 1 for all i and |a| = i a i > 2.
Definition 4.1. An a-stable n-pointed curve of genus 0 over an algebraically closed field is a tuple (C, s 1 , . . . , s n ) where C is a complete connected reduced curve C of genus 0 with at most ordinary double points ( i.e. a tree of P 1 ), s 1 , . . . , s n are closed points of C such that each s i is a regular point of C, if (s i ) i∈I coincide then i∈I a i ≤ 1, and the Q-divisor K C + a 1 s 1 + . . . + a n s n is ample, i.e. for each irreducible component C k of C we have (number of intersection points with other components) + s i ∈C k a i > 2.
Definition -Theorem 4.2. ( [Ha03]). The Hassett moduli space M 0,a is the fine moduli space of a-stable n-pointed curves of genus 0, i.e. M 0,a represents the moduli functor (denoted by the same symbol) Y → a-stable n-pointed curves of genus 0 over Y / ∼ where an a-stable n-pointed curve of genus 0 over a scheme Y is a flat proper morphism C → Y with sections s 1 , . . . , s n : Y → C such that the geometric fibres (C y , s 1 (y), . . . , s n (y)) are a-stable n-pointed curves of genus 0.

The functor of points of M 0,a
We give a description of the functor of M 0,a similar to theorem 3.9 and 3.12 based on natural embeddings of a-stable n-pointed curves of genus 0 in a product of P 1 .
Let (C → Y, s 1 , . . . , s n ) be an a-stable n-pointed curve of genus 0 over Y . Using again the methods of [Kn83], for T ⊆ {1, . . . , n}, |T | = 3 the sections (s i ) i∈T define a contraction morphism to a P 1 -bundle over Y . Restricting to the open subscheme Y T ⊆ Y where (s i ) i∈T are pairwise disjoint, we obtain a trivial P 1 -bundle P 1 Y T → Y T with sections s T 1 , . . . , s T n such that (s T i ) i∈T are pairwise disjoint. As in example 3.11 we can introduce homogeneous coordinates Let J = {i 1 , i 2 , i 3 , i 4 , i 5 } ⊆ {1, . . . , n}, |J| = 5 and I = {i 1 , i 2 , i 3 , i 4 }, I ′ = {i 1 , i 2 , i 3 , i 5 }, because over the open set {y | s i 3 (y) = s i 4 (y)} ⊆ Y T ∩ Y T ′ we have the same situation as in the case M 0,n , over {y | s i 3 (y), s i 4 (y) in same component} ⊆ Y T ∩ Y T ′ these equations are usual base changes in P 1 .
Theorem 4.3. The functor of M 0,a is isomorphic to the contravariant functor (3) ∀ T ∀ y ∈ Y T : w(y, T, a) > 2 (4) ∀ y ∈ Y ∀ a ′ ∈ Q n >0 , |a ′ | > 2, a ′ ≤ a ∃ T : y ∈ Y T , w(y, T, a ′ ) > 2 (5) ∀ T, T ′ ∀ y ∈ Y T ′ : where sets T are always subsets T ⊆ {1, . . . , n} with |T | = 3, w(y, T, a) = l min 1, i∈J l (y,T ) a i for geometric points y ∈ Y T and the partitions {1, . . . , n} = l J l (y, T ) defined by i, j ∈ J l (y, T ) for some l if and only if s T i (y) = s T j (y). We write s T i if we work with some fixed ordering of T that is not relevant.
Proof. The construction above gives a morphism from M 0,a to the functor (4.2). Of the remaining conditions (3) is satisfied because of a-stability. Concerning condition (4), for an a-stable n-pointed tree C over an algebraically closed field each intersection point p of components divides the tree into two components, where for a given a ′ ∈ Q n >0 , |a ′ | > 2, a ′ ≤ a the weights a ′ i of the marked points of at least one component sum up to > 1. Let C p be this component if it is unique, otherwise C p = C. The intersection p C p is nonempty and w(y, T, a ′ ) > 2 for each T which singles out an irreducible component of p C p . Condition (5) is satisfied because our construction defines Y T as the maximal set where the corresponding sections (s i ) i∈T are distinct.
We define the morphism in the opposite direction: for an S-scheme Y and a collection of sections s i 1 ,i 2 ,i 3 i 4 : Y T → P 1 Y T as in (4.2) we construct an a-stable n-pointed tree over Y . Let C ⊆ T P 1 Y T be the subscheme defined by the equations are homogeneous coordinates of P 1 Y T , T = {i 1 , i 2 , i 3 }, as in remark 3.13. We define sections s i = T (s T i,0 : s T i,1 ) : U → C ⊆ T P 1 Y T for i ∈ {1, . . . , n}. Let y ∈ Y be a geometric point, T y = {T | y ∈ Y T } and U ⊆ Y be an open neighbourhood of y such that ∀ T ∈ T y ∀ y ′ ∈ U : s T i (y ′ ) = s T j (y ′ ) =⇒ s T i (y) = s T j (y). Assume that s i 1 (y) = s i 2 (y), s i 2 (y) = s i ′ 2 (y), then T ′ = {i 1 , i 2 , i ′ 2 } ∈ T y . There is a T ′′ ∈ T y such that s T ′′ i 1 (y) = s T ′′ i 2 (y), s T ′′ i ′ 2 (y). Then s T ′′ i 1 (y), s T ′′ i 2 (y), s T ′′ i 3 (y) are pairwise distinct for some i 3 , so T = {i 1 , i 2 , i 3 } ∈ T y . The coordinates for T and T ′ are related over U ∩ Y T ′ by s i 1 ,i 2 ,i 3 i ′ 2 ,0 with (s i 1 ,i 2 ,i 3 i ′ 2 ,0 : s i 1 ,i 2 ,i 3 i ′ 2 ,1 ) = (0 : 1), (1 : 0) and this equation is compatible with the other equations. It follows that the projection T P 1 Y T → T =T ′ P 1 Y T induces an isomorphism of C onto the subscheme in T =T ′ P 1 Y T defined by the equations not involving T ′ . The case that 3 sections (s i ) i∈T ′ coincide over y but not over some y ′ ∈ U is similar. Thus we obtain a curve C ′ ⊆ T ∈T y P 1 U isomorphic to the original curve. Let I ⊆ {1, . . . , n} such s i (y) = s i ′ (y) for i, i ′ ∈ I and for any j ∈ {1, . . . , n} there is i ∈ I such that s j (y) = s i (y). Then, as in the above case, after restriction to factors corresponding to T ⊆ I, we have a curve C ′′ ⊆ T ⊆I P 1 U isomorphic to C ′ . By remark 3.13 this scheme C ′′ → U together with the sections (s i ) i∈I is a stable |I|-pointed tree over U .
Using conditions (3) and (4) we show that the geometric fibre (C y , s 1 (y), . . . , s n (y)) is a-stable. The property that for each irreducible component C k y of C y we have (number of intersection points with other components) + s i (y)∈C k y a i > 2 follows from condition (3). The property that if (s j (y)) j∈J coincide then a(J) = j∈J a j ≤ 1 follows from condition (4): if (s j (y)) j∈J coincide, then for any ε such that |a| − a(J) > ε > 0 there is a ′ < a such that a ′ j = a j for j ∈ J and i∈J ∁ a ′ i = max{0, 2 − a(J)} + ε, by (4) there exists T such that y ∈ Y T , w(y, T, a ′ ) > 2, and a(J) ≤ 1 follows, because choosing ε < a(J) − 1 in case a(J) > 1 would imply w(y, T, a ′ ) ≤ 2.
Thus we have shown that (C → Y, s 1 , . . . , s n ) is an a-stable n-pointed tree. This defines a morphism from the functor (4.2) to M 0,a .
One verifies that these two morphisms are mutually inverse.

Hassett moduli spaces as limits of moduli spaces of quiver representations
For a, a ′ ∈ Q n we write a ′ ≤ a if and only if a ′ i ≤ a i for all i, and we write a ′ < a if a ′ ≤ a and a ′ = a.
In theorem 3.14 we have described the Losev-Manin moduli space as the limit over moduli spaces of quiver representations M θ (P n ). By corollary 2.8 this is the same as the limit over moduli spaces of representations of the quiver Q n+2 for weights in a neighbourhood of a vertex of ∆(2, n + 2): we have L n = lim ← −θ<a M θ (Q n+2 ) for a = (1, 1, ε, . . . , ε) ∈ Q n+2 , Theorem 4.4. Let a = (a 1 , . . . , a n ) ∈ Q n such that 0 < a i ≤ 1, |a| = i a i > 2. The Hassett moduli space M 0,a is the limit over the moduli spaces of representations M θ (Q n ) for θ ∈ P (a) = {θ ∈ ∆(2, n) | θ < a}, i.e.
Proof. We verify that the results in subsection 2.4 remain valid if we consider the limit over weights in the convex polytope P (a) and obtain the following analogon of proposition 2.15: The functor lim ← −θ<a M θ (Q n ) is isomorphic to the functor Lim(Q n , a) defined by Y → (ϕ s θ ) θ ∈ θ < a generic M θ (Q)(Y ) equations (2.7) hold As in the proof of theorem 3.15 there is a description in terms of the weights θ T , but one has to take into consideration that for some (s θ ) θ≤a defining an element of Lim(Q n , a)(Y ) there might be points y ∈ Y such that the polytopes Θ(s θ (y)) for θ < a do not cover ∆(2, n) and leave out some of the θ T . So Lim(Q n , a) is isomorphic to the functor Lim ′ (Q n , a) defined by As in the proof of theorem 3.15 the equations relating s θ T , s θ T ′ for |T ∩ T ′ | = 2 are sufficient and we introduce the sections s i 1 ,i 2 ,i 3 i 4 . We obtain the functor Lim ′′ (Q n , a) defined by (3) ∀ T ∀ y ∈ Y T : int P (a) ∩ Θ(s T (y)) = ∅ (4) ∀ y ∈ Y : P (a) ⊆ y∈Y T Θ(s T (y)) (5) ∀ T, where we use s T for (s i 1 ,i 2 ,i 3 j ) j in case the order of elements of T = {i 1 , i 2 , i 3 } is irrelevant. To show that this functor coincides with functor (4.2) we compare conditions (3),(4),(5) in both. Let (J l (y, T )) l and w(y, T, a) be defined as in theorem 4.3, θ, a ′ will always be elements of Q n >0 . Concerning condition (3) for a given T and y ∈ Y T we have the equivalences int(P (a) ∩ Θ(s T (y))) = ∅ ⇐⇒ ∃ θ < a, |θ| = 2 ∀ l : i∈J l (y,T ) θ i < 1 ⇐⇒ ∃ a ′ < a : w(y, a ′ , T ) = |a ′ | > 2 ⇐⇒ w(y, a, T ) > 2.