On the invariant theory for acyclic gentle algebras

In this paper we show that the fields of rational invariants over the irreducible components of the module varieties for an acyclic gentle algebra are purely transcendental extensions. Along the way, we exhibit for such fields of rational invariants a transcendence basis in terms of Schofield determinantal semi-invariants. We also show that the moduli space of modules over a regular irreducible component is just a product of projective spaces.


INTRODUCTION
Throughout the article, k always denotes an algebraically closed field of characteristic zero. All algebras (associative and with identity) are assumed to be finite-dimensional over k, and all modules are assumed to be finite-dimensional left modules.
One of the fundamental problems in the representation theory of algebras is that of classifying the indecomposable modules. Based on the complexity of these modules, one distinguishes the class of tame algebras and that of wild algebras. According to the remarkable Tame-Wild Dichotomy Theorem of Drozd [25], these two classes of algebras are disjoint and they cover the whole class of algebras. Since the representation theory of a wild algebra is at least as complicated as that of a free algebra in two variables, and since the later theory is known to be undecidable, one can hope to meaningfully classify the indecomposable modules only for tame algebras. For more precise definitions, see [38, Chapter XIX] and the reference therein.
An interesting task in the representation theory of algebras is to study, for a given finitedimensional algebra, the geometry of the affine varieties of modules of fixed dimension vectors and the actions of the corresponding products of general linear groups. In particular, it would be interesting to find characterizations of prominent classes of tame algebras via geometric properties of their module varieties. This research direction has attracted much attention during the last two decades (see for example [4], [5], [6], [7], [12], [23] [26], [32], [33], [34], [39]).
In this paper, we seek for characterizations of tame algebras in terms of invariant theory. A first result in this direction was obtained by Skowroński and Weyman in [39,Theorem 1] where they showed that a finite-dimensional algebra of global dimension one is tame if and only if all of its algebras of semi-invariants are complete intersections. Unfortunately, this result does not extend to algebras of higher global dimension (not even of global dimension two) as shown by Kraśkiewicz in [28]. As it was suggested by Weyman, in order to characterize the tameness of an algebra via invariant theory, one should impose geometric conditions on the various moduli spaces of semi-stable modules rather than on the entire algebras of semi-invariants. For more precise details, see Section 7.
A description of the tameness of quasi-tilted algebras in terms of the invariant theory of the algebras in question has been found in [14,15]. In this paper, we continue this line of inquiry for the class of triangular gentle algebras, which are known to be tame. Their indecomposable modules can be nicely classified, however these algebras still represent an increase in the level of complexity from the tame quasi-tilted case. For example, it is possible to construct triangular gentle algebras of arbitrarily large global dimension. Furthermore, the number of one-parameter families required to parameterize d-dimensional indecomposable modules can grow faster than any polynomial in d. (1) the field of rational invariants k(C) GL(d) is a purely transcendental extension of k whose transcendece degree equals the sum of the multiplicities of the indecomposable irreducible regular components occurring in the generic decomposition of C; (2) if C is an irreducible regular component then for any weight θ ∈ Z Q 0 with C ss θ = ∅ and such that the θ-stable summands of C are regular, the moduli space M(C) ss θ is just a product of projective spaces.
Our next main result gives a transcendence basis for k(C) GL(d) when C is an irreducible regular component. We do this via the so called up and down graphs, introduced in the context of triangular gentle algebras by the first author (see [11]). The up and down graphs are combinatorially defined objects which hold valuable geometric information in that they: (1) give the generic decomposition of irreducible components in module varieties; (2) explicitly describe the generic modules in irreducible components; and (3) allow for explicit computations of the so called generalized Schofield' semi-invariants which in turn are rather remarkable coordinates on moduli spaces of modules for finitedimensional algebras. For each i, let λ(i, j), 0 ≤ j ≤ p i , be pairwise distinct elements of k * and M (d i , r i , λ(i, j)), 0 ≤ j ≤ m i , be the corresponding generic modules of mod(A, d i , r i ). Then, k(mod(A, d, r)) GL(d) is a purely transcendental extension of k of degree N = m i with transcendence basis We point out that the class of tame hereditary algebras is the only other class of tame algebras for which a similar transcendence basis has been constructed (see [35]). For wild quivers, the corresponding rationality problem, which has been open for more than 45 years, is one of the most important open problems in rational invariant theory (see [8]). It is our hope that the representation-theoretic description of the transcendence basis in the theorem above will inspire the construction of transcendence bases in other situations, too.
In Section 2, we outline the pertinent notions related to bound quiver algebras. This includes a description of the module varieties and generic decomposition of irreducible components due to Crawley-Boevey and Schröer. Section 3 recalls the basic notions of rational invariants and rational quotients. It includes a general reduction result for fields of rational invariants over irreducible components in module varieties. In Section 4, we first review King's construction of moduli spaces of modules for finite-dimensional modules, and then state another general reduction result that allows one to break a moduli space of modules into smaller ones. We also show that, at least in the tame case, these smaller moduli spaces are rather well-behaved (see Proposition 7). These general results are applied to gentle algebras in Section 5, following a recollection of the construction of their generic modules. We prove a number of facts about these irreducible components which allow an application of the reduction techniques of Sections 4 and 3. In section 6 we prove the two main theorems of the paper, and point out some idiosynchracies that arise in the generic decompositions of modules over gentle algebras. Section 7 concludes the article by placing the results in the framework of an effort to characterize tameness of an algebra via invariant theoretic characteristics.
2.1. Bound quiver algebras. Let Q = (Q 0 , Q 1 , t, h) be a finite quiver with vertex set Q 0 and arrow set Q 1 . The two functions t, h : Q 1 → Q 0 assign to each arrow a ∈ Q 1 its tail ta and head ha, respectively.
A representation V of Q over k is a collection (V (i), V (a)) i∈Q 0 ,a∈Q 1 of finite-dimensional k-vector spaces V (i), i ∈ Q 0 , and k-linear maps V (a) ∈ Hom k (V (ta), V (ha)), a ∈ Q 1 . The dimension vector of a representation V of Q is the function dim V : Q 0 → Z defined by (dim V )(i) = dim k V (i) for i ∈ Q 0 . Let S i be the one-dimensional representation of Q at vertex i ∈ Q 0 . By a dimension vector of Q, we simply mean a vector d ∈ Z Q 0 ≥0 .
Given two representations V and W of Q, we define a morphism ϕ : V → W to be a collection (ϕ(i)) i∈Q 0 of k-linear maps with ϕ(i) ∈ Hom k (V (i), W (i)) for each i ∈ Q 0 , and such that ϕ(ha)V (a) = W (a)ϕ(ta) for each a ∈ Q 1 . We denote by Hom Q (V, W ) the k-vector space of all morphisms from V to W . Let V and W be two representations of Q. We say that V is a subrepresentation of W if V (i) is a subspace of W (i) for each i ∈ Q 0 and V (a) is the restriction of W (a) to V (ta) for each a ∈ Q 1 . In this way, we obtain the abelian category rep(Q) of all representations of Q.
Given a quiver Q, its path algebra kQ has a k-basis consisting of all paths (including the trivial ones) and the multiplication in kQ is given by concatenation of paths. It is easy to see that any kQ-module defines a representation of Q, and vice-versa. Furthermore, the category mod(kQ) of kQ-modules is equivalent to the category rep(Q). In what follows, we identify mod(kQ) and rep(Q), and use the same notation for a module and the corresponding representation.
A two-sided ideal I of kQ is said to be admissible if there exists an integer L ≥ 2 such that R L Q ⊆ I ⊆ R 2 Q . Here, R Q denotes the two-sided ideal of kQ generated by all arrows of Q.
If I is an admissible ideal of kQ, the pair (Q, I) is called a bound quiver and the quotient algebra kQ/I is called the bound quiver algebra of (Q, I). Any admissible ideal is generated by finitely many admissible relations, and any bound quiver algebra is finite-dimensional and basic. Moreover, a bound quiver algebra kQ/I is connected if and only if (the underlying graph of) Q is connected (see for example [2,Lemma II.2.5]).
Up to Morita equivalence, any finite-dimensional algebra A can be viewed as the bound quiver algebra of a bound quiver (Q A , I), where Q A is the Gabriel quiver of A (see [2, Corollary I.6.10 and Theorem II.3.7]). (Note that the ideal of relations I is not uniquely determined by A.) We say that A is a triangular algebra if its Gabriel quiver has no oriented cycles.
Fix a bound quiver (Q, I) and let A = kQ/I be its bound quiver algebra. A representation M of A (or (Q, I)) is just a representation M of Q such that M (r) = 0 for all r ∈ I. The category mod(A) of finite-dimensional left A-modules is equivalent to the category rep(A) of representations of A. As before, we identify mod(A) and rep(A), and make no distinction between A-modules and representations of A. For each vertex x ∈ Q 0 , we denote by P x the projective indecomposable A-module at vertex x. For an A-module M , we denote by pdimM its projective dimension. An A-module is called Schur if End A (M ) ∼ = k.
An algebra A is called tame if for each dimension vector d ∈ Z Q 0 ≥0 , the subcategory ind d (A), whose objects are the indecomposable d-dimensional A-modules, is parametrized, the sense of [38], by a finite family of functors (2) Q i is an A−A i -bimodule which is finitely generated and free as a right A i -module; and (3) the functor F i is a representation embedding. For more details, we refer to [24] and [38].
For the remainder of this subsection, we assume that A has finite global dimension; this happens, for example, when Q has no oriented cycles. The Euler form of A is the bilinear form ·, · A : by simultaneous conjugation, i.e., for g = (g(i)) i∈Q 0 ∈ GL(d) and V = (V (a)) a∈Q 1 ∈ mod(Q, d), g · V is defined by It can be easily seen that mod(A, d) is a GL(d)-invariant closed subvariety of mod(Q, d), and that the GL(d)−orbits in mod(A, d) are in one-to-one correspondence with the isomorphism classes of the d-dimensional A-modules. Note that mod(A, d) does not have to be irreducible.
Let C be an irreducible component of mod(A, d). We say that C is indecomposable if C has a non-empty open subset of indecomposable modules; whenever mod(A, d) has such an irreducible component, we say that d is a generic root of A.
we denote by C 1 ⊕ . . . ⊕ C t the constructible subset of mod(A, d) defined as: It also follows from [17, For the remainder of this section, we assume that A is a tame algebra and let d be a generic root of A. Denote by ind(A, d) the constructible subset of mod(A, d) consisting of all d-dimensional indecomposable A-modules.
We know from the work of Dowbor and Skowroński in [24] that there are finitely many principal open subsets U i ⊆ A 1 = k and regular morphisms f i : Consequently, we have that . From the discussion above, it follows that either: In the tame case, we have the following simple but very useful dimension count: Lemma 3. Let A = kQ/I be a tame bound quiver algebra, d a generic root, and C ⊆ mod(A, d) an indecomposable irreducible component. Then, Proof. If C is an orbit closure then the expression above is obviously zero. The only other possibility is when C = λ∈U GL(d)f (λ) with (U, f ) a parameterizing pair as above. The action morphism µ : GL(d) × U → C, µ(g, λ) = gf (λ), ∀(g, λ) ∈ GL(d) × U, is dominant by construction and, moreover, for a generic M 0 = µ(g 0 , λ 0 ) ∈ C: In particular, this shows that dim µ −1 (M 0 ) = dim k End A (M 0 ). Using now the theorem on the generic fiber, we get that dim GL(d) which is exactly what we need to prove.

RATIONAL INVARIANTS
Let A = kQ/I be a bound quiver algebra, d ∈ Z Q 0 ≥0 a dimension vector of A, and C a GL(d)-invariant irreducible closed subset of mod(A, d).
The field of rational GL(d)-invariants on C, denoted by k(C) GL(d) , is defined as follows: Our motivation for studying such fields of rational invariants is twofold: on one hand, k(C) GL(d) is the function field of the rational quotient (in the sense of Rosenlicht) of C by GL(d) which parametrizes the generic A-modules in C. Hence, these fields play a key role in the generic representation theory of A.
On the other hand, there are important projective (parameterizing) varieties, such as the Hilbert scheme of points on P 2 (see [1]) or various moduli spaces of sheaves over rational surfaces (see [37]), whose function fields can be viewed as fields of rational invariants for bound quiver algebras. Thus, it is desirable to know how methods and ideas from representation theory of algebras can be used to shed light on the rationality question for such varieties.
In what follows, we explain how to reduce the problem of describing fields of rational invariants on irreducible components of module varieties to the case where the irreducible components involved are indecomposable. But first we recall some fundamental facts from rational invariant theory for which we refer the reader to [31] and the reference therein. Let G be a linear algebraic group acting regularly on an irreducible variety X. The field k(X) G of G-invariant rational functions on X is always finitely generated over k since it is a subfield of k(X) which is finitely generated over k. A rational quotient of X by (the action of) G is an irreducible variety Y such that k(Y ) = k(X) G together with the dominant rational map π : X Y induced by the inclusion k(X) G ⊂ k(X). In [36], Rosenlicht shows that there is always a rational quotient of X by G, which is uniquely defined up to birational isomorphism. In fact, it is proved that there is a G-invariant open and dense subset X 0 of X such that the restriction of π to X 0 is a dominant regular morphism and π −1 (π(x)) = Gx for all x ∈ X 0 (see [36,Theorem 2] or [9, Section 1.6]). Furthermore, one can show that a rational quotient π : X Y satisfies the following universal property (see [ Y be a rational map such that ρ −1 (ρ(x)) = Gx for x ∈ X in general position. Then there exists a rational map ρ : Y Y such that ρ = ρ • π. If in addition ρ is dominant then ρ becomes a birational isomorphism. One usually writes X/G in place of Y and call it the rational quotient of X by G.
Remark 1. From the discussion above and using the theorem on the generic fiber, one can immediately see that This formula combined with Lemma 3 shows that for a tame algeba A, a dimension vector d, and an indecomposable irreducible In what follows, if R is an integral domain, we denote its field of fractions by Quot(R). Moreover, if K/k is a field extension and m is a positive integer, we define S m (K/k) to be the field (Quot(K ⊗m )) Sm which is, in fact, the same as Quot((K ⊗m ) Sm ) since S m is a finite group. Now we are ready to prove the following reduction result (compare to [14,Proposition 4.7]): . . , C l orbit closures, m 1 , . . . , m n are positive integers, and Proof. Note that we can write The proof will follow from the lemma below.
Lemma 5. Assume that C ⊆ mod(A, d) can be decomposed as Then, it is easy to see that S is an irreducible H-invariant closed subvariety of C such that: (1) GS = C; (2) for a generic s ∈ S, we have Gs ∩ S = Hs. Now, let π : C C/G be the rational quotient of C by G. Then, (1) ensures that the restriction ρ of π to S is a well-defined dominant rational map, and (2) simply says that for generic s ∈ S, ρ −1 (ρ(s)) = Hs. It follows from the universal property for rational quotients that ρ is the rational quotient of S by H, and so k(

Remark 2.
This lemma tells us that, for the purposes of computing rational invariants, we can always get rid of the orbit closures that occur in a generic decomposition. If, additionally, the other irreducible components that occur in a generic decomposition can 8 be separated as in Proposition 4 then we have a further reduction in the fields of rational invariants.

MODULI SPACES OF MODULES
Let A = kQ/I be a bound quiver algebra and let d ∈ Z Q 0 ≥0 be a dimension vector of A. The ring of invariants I(A, d) := k[mod(A, d)] GL(d) turns out to be precisely the base field k since A is finite-dimensional. However, the action of the subgroup SL(d) ⊆ GL(d), defined by provides us with a highly non-trivial ring of semi-invariants. Note that any θ ∈ Z Q 0 defines a rational character χ θ : GL(d) → k * by In this way, we can identify Γ = Z Q 0 with the group X (GL(d)) of rational characters of GL(d), assuming that d is a sincere dimension vector. In general, we have only the natural epimorphism Γ → X * (GL(d)). We also refer to the rational characters of GL(d) as (integral) weights of A (or Q).
Let   We will prove in Section 5 that the same is true for triangular gentle algebras.
Let C be a θ-well-behaved irreducible component of mod(A, d). We say that • the generic A-module M in C has a finite filtration 0 = M 0 ⊆ M 1 ⊆ · · · ⊆ M l = M of submodules such that each factor M j /M j−1 , 1 ≤ j ≤ l, is isomorphic to a θ-stable module in one the C 1 , . . . , C l , and the sequence (dim M 1 /M 0 , . . . , dim M/M l−1 ) is the same as (d 1 , . . . , d l ) up to permutation. We call C 1 , . . . , C l the θ-stable summands of C. To prove the existence and uniqueness of the θ-stable decomposition of C, first note that the irreducible variety C ss θ is a disjoint union of sets of the form F (C i ) 1≤i≤l , where each F (C i ) 1≤i≤l consists of those modules M ∈ C that have a finite filtration 0 = M 0 ⊆ M 1 ⊆ · · · ⊆ M l = M of submodules with each factor M j /M j−1 isomorphic to a θ-stable module in one the C i , 1 ≤ i ≤ l. (Note that the θ-well-behavedness of C is used to ensure that the union above is indeed disjoint.) Next, it is not difficult to show that each F (C i ) 1≤i≤l is constructible (see for example [17,Section 3]). Hence, there is a unique (up to permutation) sequence (C i ) 1≤i≤l of θ-stable irreducible components for which F (C i ) 1≤i≤l contains an open and dense subset of C ss θ (or C). Now, we are ready to state the following reduction result from [15, Theorem 1.4]): Theorem 6. Let A = kQ/I be a bound quiver algebra and let C ⊆ mod(A, d) be a θ-well-behaved irreducible component where θ is an integral weight of A. Let C = m 1 · C 1 . . . m n · C n be the θ-stable decomposition of C where C i ⊆ mod(A, d i ), 1 ≤ i ≤ n, are θ-stable irreducible components and d i = d j for all 1 ≤ i = j ≤ n. Assume that C is a normal variety and . Note that this reduction result allows us to "break" a moduli space of modules into smaller ones. We show next that these smaller moduli spaces are rather well behaved, at least in the tame case. Proposition 7. Let A = kQ/I be a tame bound quiver algebra, d a generic root of A, and C ⊆ mod(A, d) an indecomposable irreducible component. Then, the following statements hold.
(1) For any weight θ ∈ Z Q 0 with C ss θ = ∅, M(C) ss θ is either a point or projective curve. (2) If θ ∈ Z Q 0 is so that C s θ = ∅ then M(C) ss θ is rational. If, in addition, C is normal then M(C) ss θ is either a point or P 1 . (3) If C is a Schur component then k(C) GL(d) is a rational field of transcendence degree at most one.
Proof. If C is an obit closure then (1) − (3) are obviously true. Otherwise, we have seen in where (U ⊆ k * , f : U → C) is a parameterizing pair.
(1) Let θ ∈ Z Q 0 be an integral weight and π : C ss θ → M(C) ss θ be the quotient morphism. Then, for a generic module M 0 ∈ C ss θ , we have: which is at most 1 by Lemma 3. This proves (1).
The injectivity of ϕ together with the fact that dim U 0 = 1 and dim M(C) ss θ ≤ 1 implies that ϕ is an injective dominant morphism, and hence is birational. This shows that M(C) ss θ is a rational projective curve. If, in addition, C is normal then so is any good quotient of C. In particular, under this extra assumption, M(C) ss θ is a rational normal projective curve, i.e. M(C) ss θ is precisely P 1 .
(3) Notice that {λ ∈ U | dim k End A (f (λ)) = min{dim k End A (M ) | M ∈ C}} is a nonempty open subset of U. From this we deduce that C contains infinitely many, pairwise non-isomorphic, Schur A-modules. Hence, C must contain a homogenous Schur A-module M by [18,Theorem D]. It has been proved in [16,Lemma 11] that M is θ Mstable where θ M is a specific integral weight associated to M (see also Section 6). Using Remark 3 and part (2), we can now see that

Remark 5.
As a consequence of the proposition above and Theorem 6, we obtain that for a tame algebra A and an irreducible component C ⊆ mod(A, d) for which Theorem 6 is applicable, the moduli space M(C) ss θ is a rational variety.

ACYCLIC GENTLE ALGEBRAS
Gentle algebras are a particularly well-behaved class of string algebras that have recently enjoyed a resurgence in popularity, primarily due to their appearance in the study of cluster algebras arising from unpunctured surfaces (see [3]). String algebras are (nonhereditary) tame algebras whose indecomposable modules can be parameterized by certain walks on the underlying quiver Q not passing through the ideal I, and whose irreducible morphisms can be described by operations on these walks [10]. The invariant theory for a particular class of triangular gentle algebras was first studied in detail by Kraśkiewicz and Weyman [29], and then by the second author in [12,11,13]. In these latter works, the irreducible components of triangular gentle algebras are determined. Their rings of semi-invariants are shown to be semigroup rings, and the generic modules in irreducible components are constructed. This allowed for the calculation of the quotients M(C) ss θ for some special choices of C and θ. A bound quiver algebra kQ/I is called a gentle algebra if the following properties hold: (1) for each vertex i ∈ Q 0 there are at most two arrows with head i, and at most two arrows with tail i; (2) for any arrow b ∈ Q 1 , there is at most one arrow a ∈ Q 1 and at most one arrow c ∈ Q 1 such that ab / ∈ I and bc / ∈ I; (3) for each arrow b ∈ Q 1 there is at most one arrow a ∈ Q 1 with ta = hb (resp. at most one arrow c ∈ Q 1 with hc = tb) such that ab ∈ I (resp. bc ∈ I); (4) I is generated by paths of length 2. The bound quiver algebra kQ/I is called a gentle algebra.
In [12], colorings of a quiver were introduced to understand the module varieties of triangular gentle algebras. A coloring c of a quiver Q is a surjective map c : Q 1 → S, where S is some finite set whose elements we call colors, such that c −1 (s) is a directed path for all s ∈ S. Given a coloring c of Q, define the coloring ideal I c to be the two-sided ideal in kQ generated by monochromatic paths of length two, i.e. I c = ba | c(b) = c(a) and ha = tb .  d(i)) when either b or a fails to exist). The rank function r is called maximal if it is so under the coordinate-wise partial order (namely r ≤ r if and only if r (a) ≤ r(a) for all a ∈ Q 1 ). Note that a rank function r defines a closed subvariety of mod(A, d): It was shown in [12] that every irreducible component of mod (A, d) is of the form mod (A, d, r) for r maximal. These varieties can then be viewed as products of varieties of complexes taken along each colored path. DeConcini-Strickland [19] showed that varieties of complexes are normal, and thus mod(A, d, r) is normal for any choice of d, r.

Up and down graphs.
We now focus exclusively on the case in which A = kQ/I c is a triangular gentle algebra. Under this restriction, each vertex has at most two colors incident to it. Denote by X the set of pairs (i, s) ∈ Q 0 × S such that s is a color incident to i. A sign function is a map : X → {±1} satisfying the property that (i, The up and down graph Γ(Q, c, d, r, ) is the directed graph with vertices {v i j | i ∈ Q 0 , j = 1, . . . , d(i)} and arrows {f a j | a ∈ Q 1 , j = 1, . . . , r(a)} so that There is an obvious morphism of quivers π : Γ(Q, c, d, r, ) → Q with π(v i j ) = i and π(f a j ) = a. This, in turn, gives rise to a pushforward map π * : rep(Γ(Q, c, d, r, Notice that since r is a rank function, by definition of Γ(Q, c, d, r, ), π * (V )(b)•π * (V )(a) = 0 whenever ha = tb and c(a) = c(b). Therefore the image of π * lies in mod(A). Furthermore, each vertex in Γ is incident to at most two arrows, and so the connected components of Γ(Q, c, d, r, ) are either chains or cycles (components which we refer to as strings and bands, respectively). Let B be the set of band components in Γ, and for b ∈ B choose a ver- The up and down module M (Q, c, d, r, , λ) is defined to be π * ( M (Q, c, d, r, , λ)). When the quiver Q and coloring c are understood from context, we will simply write Γ(d, r, ) and M (d, r, , λ).

Example 1.
Consider the following bound quiver (Q, I c ) with the coloring as indicated by the type of arrow.
We consider the dimension vector d and rank function r indicated in the diagram below, with dimensions in boxes and ranks as decorations of the arrows.
Take as indicated, where we place (x, c(a)) on the arrow a near the vertex x: The up and down graph Γ(d, r, ) consists of one band component and one string component. One possible choice of Θ(b) for the unique band is the vertex framed by a circle 14 here: T T n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n v (4) 1 Then M (d, r, , λ) is the module given by In [11], an explicit minimal projective resolution for each M (d, r, , λ) is constructed. Using this resolution it is possible to show the following: , r, , λ), M (d, r, , λ )) = 0 whenever λ, λ ∈ (k * ) B are vectors that share no common coordinates, and that when Γ(d, r, ) consists of a single band component, , r, , λ), M (d, r, , λ)) = k. 15 In particular, these results can be exploited to show that When there are no band modules, then, mod(A, d, r) is simply the orbit closure of the rigid module M (d, r, ). This means that in order to apply the results of Section 3, we need only understand those irreducible components whose generic modules are direct sums of band modules. Given these results, we notice that the family GL(d)M (d, r, , λ) is independent of the choice of . In the subsequent sections we will suppress this variable when referring to both the up-and-down graph and the up-and-down modules.
It is important to note that the indecomposable direct summands of M (d, r, , λ) correspond to connected components of Γ(d, r, ). Thus, the graph Γ(d, r, ) also gives the generic decomposition of mod(A, d, r), i.e., the dimension vectors d i and rank functions r i such that where mod(A, d i , r i ) are indecomposable irreducible components. In particular, in order to apply the reduction techniques from Proposition 4 and Theorem 6, it needs to be shown that if mod(A, d i , r i ) and mod(A, d j , r i ) are direct summands that are not orbit closures, and d i = d j , then r i = r j .

Regular Irreducible Components. An irreducible component mod(A, d, r)
is called regular if the generic module M (d, r, λ) is a direct sum of bands. That is the only connected components of Γ(d, r) are cycles. We can characterize the ranks for which this is the case in the following proposition. There are a number of important consequences that follow from this combinatorial requirement. Proof. Suppose that mod(A, d) contains an irreducible regular component. We will show that r is determined uniquely by the dimension vector and Proposition 9. Given any color s ∈ S, let a ns · . . . · a 1 be the full path of the elements c −1 (s). Let i j = ha j and i 0 = ta 1 . Then r(a 1 ) = d(i 0 ), and r(a j+1 ) = d(i j ) − r(a j ), so the rank function r for which mod(A, d, r) is regular is uniquely determined by d. As for the Euler form, one simply expresses the matrix of the form as a sum of contributions along each color, and then uses the fact that if mod(A, d, r) is regular, then for any fully colored path as above, ns j=0 (−1) j d(i j ) = 0. Precise details can be found in [13,Chapter 5.1].
It will be convenient to describe the projective resolution for X λ := M (d, r, λ) when mod(A, d, r) is a regular irreducible component (the general case is handled in [11,Section 3]). This will allow us to derive a closed formula for the Schofield semi-invariants of weight d, − . Let us denote by S 0 and S 1 the set of sources and sinks of Γ = Γ(d, r), respectively. For any v x i ∈ S 1 , we distinguish two distinct paths in Γ terminating at v x i , denoted l + (v x i ) and l − (v x i ). The path l δ (v x i ) is determined by the following two conditions: Notice that this indeed defines two distinct paths, since in Γ each vertex is incident to precisely two colored arrows whose signs under are different. We now consider the modules with a map F : P 1 → P 0 defined in the following way.
) be the map given by Following F | (i,x) by the inclusion gives a map P (v x i ) → P 0 which we denote with the same symbol. Then F is defined to be the map v x i ∈S 1 F | (i,x) . In [11] it was shown that P 1 → P 0 → X λ → 0 is a minimal projective resolution of X λ . In particular, X λ has projective dimension one for all λ ∈ (k * ) B . One important consequence is the following. Proof. Part 1 follows immediately from proposition above and Proposition 7(3). As for part 2, we have already mentioned that mod(A, d, r) is normal, so M(mod(A, d, r)) ss θ is either a point or P 1 by Proposition 7 (2). But from part (1) and Remark 3, the moduli space is not a point.
It turns out that we can actually extend the second part of the previous result to the case of triangular string algebras: Corollary 13. Let A = kQ/I be a triangular string algebra, d a generic root of A, and C ⊆ mod(A, d) an indecomposable irreducible component. If θ ∈ Z Q 0 is an integral weight such that C s θ = ∅ then M(C) ss θ is either a point or P 1 . Proof. It has been proved in [12,Proposition 2.8] that there exists a coloring c of Q such that I c ⊆ I and A = kQ/I c is a colored gentle algebra. Hence, there exists an irreducible component C ⊆ (A , d) that contains C.
If C is an orbit closure then the moduli space in question is just a point. The other case left is when M(C) ss θ is a rational projective curve (see Proposition 7 (2)). Then, M(C ) ss θ P 1 by Corollary 12. Now, let π : C ss θ → M(C) ss θ and π : C ss θ → M(C ) ss θ be the quotient morphisms for the actions of PGL(d) on C ss θ and C ss θ , respectively. Now, consider the PGL(d)-invariant morphism ϕ : C ss θ → M(C ) ss θ that sends M to π (M ). From the universal property of GIT quotients, we know that there exists a morphism f : It is easy to see that f is injective: Indeed, let M 1 , M 2 ∈ C ss θ be so that f (π(M 1 )) = f (π(M 2 )). Then, π (M 1 ) = π (M 2 ), which is equivalent to GL(d)M 1 ∩ GL(d)M 2 ∩ C ss θ = ∅. Observe that C ss θ is closed in C ss θ and hence GL(d)M i ∩ C ss θ , i = 1, 2, are contained in C ss θ . So, GL(d)M 1 ∩ GL(d)M 2 ∩ C ss θ = ∅, which is equivalent to π(M 1 ) = π(M 2 ). Moreover, it is clear now that f is surjective since otherwise Im f would be just a point.
In conclusion, f is a bijective morphism whose target variety is normal, and hence it has to be an isomorphism by (a consequence of) the Zariski's Main Theorem (in characteristic zero).

PROOFS OF THE MAIN RESULTS
We first prove Theorem 1, and then proceed with the proof of our constructive Theorem 2.
Proof of Theorem 1. (1) Using Corollary 10, we know that we can write the generic decomposition of C as: But note that for each 1 ≤ i ≤ n, k(C i ) GL(d i ) k(t) by Corollary 12 (1). Applying now Proposition 4, we obtain that k(C) GL(d) k(t 1 , . . . , t N ) where N = n i=1 m i .
(2) First of all, any θ-semi-stable irreducible component C ⊆ mod(A, d) is θ-well-behaved. This follows immediately from the uniqueness property in Corollary 10. Now, let us assume that C is a θ-semi-stable irreducible regular component whose θstable decomposition is of the form: . . , C l indecomposable irreducible regular components. Using Corollary 10 again, we can write the θ-stable decomposition of C as: To prove Theorem 2, we will use the projective resolution from Section 5.2. But first let us briefly recall the construction of the so-called generalized Schofield semi-invariants on module varieties following Derksen and Weyman [21], and Domokos [22].
Let X be an A-module and P 1 F − → P 0 → X → 0 be a fixed minimal projective presentation of X in mod(A). Let θ X ∈ Z Q 0 be the integral weight defined so that θ X (v) is the multiplicity of P v in P 0 minus the multiplicity of P v in P 1 for all v ∈ Q 0 . For an arbitrary A-module M , we have θ X (dim M ) = dim k Hom A (P 0 , M ) − dim k Hom A (P 1 , M ) = dim k Hom A (X, M ) − dim k Hom A (M, τ X).
Here, τ X is the Auslander-Reiten translation of X given by D(Coker f t ) where (−) t = Hom A (−, A) and D = Hom k (−, k) (for more details, see [2,§IV.2]). Notice that if pdimX ≤ 1 then θ X (dim M ) = dim k Hom A (X, M ) − dim k Ext 1 A (X, M ) = dim X, dim M . Before we continue our discussion on Schofield semi-invariants we mention the following example of a weight that satisfies the conditions in Theorem 1(2): Example 2. Let C ⊆ mod(A, d) be an irreducible regular component. Then, it is generic decomposition is of the form: where C i ⊆ mod(A, d i ), 1 ≤ i ≤ m, are indecomposable irreducible regular components and d i = d j , ∀1 ≤ i = j ≤ m. Notice also that d i , d i = 0 for all 1 ≤ i, j ≤ m (for details, see Lemma 15).
We claim that the weight θ := d, · satisfies the conditions in Theorem 1 (2). Specifically, we claim that the θ-stable decomposition of C is precisely C = l 1 · C 1 . . . l m · C m .
For this, it is clearly enough to show that C s i,θ = ∅ for each 1 ≤ i ≤ m. Let M i ∈ C i be a Schur homogeneous module (e.g. choose M i to be any indecomposable band module in C i ). Let us check that M i ∈ C s i,θ . Set M := m i=1 M l i i and note that pdimM ≤ 1 and τ M M since each M i has these two properties. From the discussion above, we have that θ = θ M and so This shows that C has the desired θ-stable decomposition. We can now apply Theorem 1(2) and conclude that M(C) ss θ m i=1 P l i . Let d ∈ Z Q 0 be a dimension vector of A such that θ X (d) = 0. Then for any module M ∈ mod(A, d), dim k Hom A (P 0 , M ) = dim k Hom A (P 1 , M ) and hence the linear map It is easy to see that c X is a semi-invariant of weight θ X . Moreover, any other choice of a minimal projective presentation of X leads to the same semi-invariant, up to a non-zero scalar. We call c X a generalized Schofield semi-invariant.
We can now apply this setup to triangular gentle algebras A = kQ/I c , when X = M (d, r, λ), a generic indecomposable regular module. Let