MODULI SPACES OF REPRESENTATIONS OF SPECIAL BISERIAL ALGEBRAS

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1 MODULI SPACES OF REPRESENTATIONS OF SPECIAL BISERIAL ALGEBRAS ANDREW T. CARROLL, CALIN CHINDRIS, RYAN KINSER, AND JERZY WEYMAN ABSTRACT. We show that the irreducible components of any moduli space of semistable representations of a special biserial algebra are always isomorphic to products of projective spaces of various dimensions. This is done by showing that certain varieties of representations of special biserial algebras are isomorphic to products of varieties of circular complexes, and therefore normal, allowing us to apply recent results of the second and third authors on moduli spaces. CONTENTS. Introduction 2. Representation varieties and moduli spaces 2 3. Varieties of circular complexes 5 4. Representation varieties of special biserial algebras and proof of the main result 8 References 2. INTRODUCTION Throughout, K denotes an algebraically closed field of characteristic zero. Unless otherwise specified, all quivers are assumed to be finite and connected, and all algebras are assumed to be bound quiver algebras. In this paper, we study representations of algebras within the general framework of Geometric Invariant Theory (GIT). This interaction between representations of algebras and GIT leads to the construction of moduli spaces of representations as solutions to the classification problem of semi-stable representations, up to S-equivalence. We point out that these moduli spaces can be arbitrarily complicated; indeed, arbitrary projective varieties can arise as moduli spaces of representations of algebras [Hil96, HZ98]. Our goal in this paper is to understand these moduli spaces for special biserial algebras. The results we obtain here are, in fact, part of a program aimed at finding geometric characterizations of the representation type of bound quiver algebras. This line of research has attracted a lot of attention, see for example [BC09, BCHZ5, Bob08, BS99, Bob4, Bob2, CW3, Chi09, Chi, CKW5, CC5a, CK6, Dom, GS03, Rie04, RZ04, RZ08, SW00]. Special biserial algebras play a prominent role in the representation theory of algebras and related areas. Their indecomposable representations can be nicely described, however the number of -parameter families needed to parametrize the n-dimensional 200 Mathematics Subject Classification. 6G20, 4D20. Key words and phrases. gentle algebras, moduli spaces, representations, regular irreducible components, special biserial algebras, varieties of circular complexes. C.C. was supported by NSA grant H and J.W. by NSF grant DMS

2 indecomposables can grow faster than any polynomial in n. Furthermore, gentle algebras and Brauer graph algebras, which are particular cases of special biserial algebras, have recently played an important role in the study of Jacobian and cluster algebras, see for example [ABCJP0, GLFS6, MS4]. Our main result is the following theorem which classifies the irreducible components of moduli spaces for special biserial algebras. Theorem. Let A be a special biserial algebra. Then any irreducible component of a moduli space M(A, d) ss is isomorphic to a product of projective spaces. The isomorphism of the theorem results from a general decomposition theorem for moduli spaces proved in [CK7]. The key geometric condition needed to apply this theorem is that certain representation varieties are normal. In this paper, we show in Proposition 9 that this condition holds in all cases relevant to special biserial algebra by reducing the consideration to varieties of circular complexes (see Sections 3 and 4). Acknowledgements. This project began during a visit of the first three authors to the University of Connecticut. The authors would like to acknowledge the generous support of the Stuart and Joan Sidney Professorship of Mathematics endowment for making the visit possible. We also thank Amelie Schreiber for participating in discussions about the project and Corrado de Concini for inspiring conversations. 2. REPRESENTATION VARIETIES AND MODULI SPACES 2.. Representation varieties. According to Gabriel s theorem, any finite-dimensional unital, associative K-algebra A can be viewed as a bound quiver algebra, up to Morita equivalence; that is there exist a quiver Q, uniquely determined by A, and an admissible ideal I of KQ such that A KQ/I. Throughout, we will adopt the language of representations of bound quivers. In particular, by abuse of terminology, we refer to a representation of Q satisfying the relations in I as a representation of A. Whenever we work with a set of generators R for I, we will always assume each generator is a linear combination of paths with the same source and target vertex. We write Q 0 for the set of vertices of Q, and Q for its set of arrows. For a dimension vector d Z Q 0 0, the affine representation variety rep(a, d) parametrizes the d-dimensional representations of (Q, R) along with a fixed basis. Writing ta and ha for the tail and head of an arrow a Q, we have: rep(a, d) := {M Mat d(ha) d(ta) (K) M(r) = 0, for all r R}. a Q Under the action of the change of base group GL(d) := x Q 0 GL(d(x), K), the orbits in rep(a, d) are in one-to-one correspondence with the isomorphism classes of d-dimensional representations of (Q, R). For more background on representation varieties, see [Bon98, Zwa]. In general, rep(a, d) does not have to be irreducible. Let C be an irreducible component of rep(a, d). We say that C is indecomposable if C has a non-empty open subset of indecomposable representations. We say that C is a Schur component if C contains a Schur representation, in which case C has a non-empty open subset of Schur representations; in particular, any Schur component is indecomposable. As shown by de la Peña in [dlp9] 2

3 and Crawley-Boevey and Schröer in [CBS02, Theorem.] any irreducible component C rep(a, d) satisfies a Krull-Schmidt type decomposition C = C... C l for some indecomposable irreducible components C i rep(a, d i ) with d i = d. Moreover, C,..., C l are uniquely determined by this property Semi-Invariants. Let A = KQ/I be an algebra and d Z Q 0 0 a dimension vector of A. We are interested in the action of SL(d) := x Q 0 SL(d(x), K) on the representation variety rep(a, d). The resulting ring of semi-invariants SI(A, d) := K[rep(A, d)] SL(d) has a weight space decomposition over the group X (GL(d)) of rational characters of GL(d): SI(A, d) = SI(A, d) χ. For each character χ X (GL(d)), χ X (GL(d)) SI(A, d) χ = {f K[rep(A, d)] g f = χ(g)f for all g GL(d)} is called the space of semi-invariants on rep(a, d) of weight χ. For a GL(d)-invariant closed subvariety C rep(a, d), we similarly define the ring of semi-invariants SI(C) := K[C] SL(d), and the space SI(C) χ of semi-invariants of weight χ X (GL(d)). Note that any Z Q 0 defines a rational character χ : GL(d) K by () χ ((g(x)) x Q0 ) = x Q 0 det g(x) (x). In this way, we get a natural epimorphism Z Q 0 X (GL(d)); we refer to the rational characters of GL(d) as integral weights of Q (or A). In case d is a sincere dimension vector, this epimorphism is an isomorphism which allows us to identify Z Q 0 with X (GL(d)) Moduli spaces of representations. Let (Q, R) be a bound quiver, and Z Q 0 an integral weight of Q. Following King [Kin94], a representation M of (Q, R) is said to be -semi-stable if (dim M) = 0 and (dim M ) 0 for all subrepresentations M M. We say that M is -stable if M is non-zero, (dim M) = 0, and (dim M ) < 0 for all subrepresentations 0 M < M. Finally, we call M a -polystable representation if M is a direct sum of -stable representations. Now, let d be a dimension vector of (Q, R) and consider the (possibly empty) open subsets rep(a, d) ss = {M rep(a, d) M is -semi-stable} and rep(a, d) s = {M rep(a, d) M is -stable} of d-dimensional (-semi)-stable representations of (Q, R). Using methods from Geometric Invariant Theory, King shows in [Kin94] that the projective variety ( ) M(A, d) ss := Proj SI(A, d) n 3 n 0

4 is a GIT-quotient of rep(a, d) ss by the action of PGL(d) where PGL(d) = GL(d)/T and T = {(λ Id d(x) ) x Q0 λ k } GL(d). Moreover, there is a (possibly empty) open subset M(A, d) s of M(A, d)ss which is a geometric quotient of rep(a, d) s by PGL(d). We say that d is a -(semi-)stable dimension vector of A if rep(a, d) (s)s. For a given GL(d)-invariant closed subvariety C of rep(a, d), we similarly define C ss, Cs, M(C) ss, and M(C)s. We say that C is a -(semi-)stable subvariety if C(s)s. From now on, let us assume that the character χ X (GL(d)) induced by is not trivial, i.e. the restriction of to the support of d is not zero, and denote by G the kernel of χ. Let C be a -semi-stable GL(d)-invariant, irreducible, closed subvariety of rep(a, d). Then we have that: K[C] G = SI(C) n. n 0 The restriction homomorphism K[rep(A, d)] K[C] remains surjective after taking G - invariants since G is linearly reductive in characteristic zero. This surjective homomorphism K[rep(A, d)] G K[C] G of graded algebras gives rise to a closed embedding M(C) ss M(A, d) ss. In fact, the image of this embedding is precisely π(css ), where π : rep(a, d) ss M(A, d)ss is the quotient morphism. The points of M(C) ss correspond bijectively to the (isomorphism classes of) -polystable representations in C. Indeed, each fiber of π : C ss M(C) ss contains a unique closed GL(d)-orbit in C ss. On the other hand, as proved by King in [Kin94, Proposition 3.2(i)], these orbits are precisely the isomorphism classes of -polystable representation in C. In fact, for any M C ss, there exists a -psg λ X (G ) such that M := lim t 0 λ(t)m exists and is the unique, up to isomorphism, polystable representation in GL(d)M C ss. The goal now is to explain how to decompose a given irreducible component of a moduli space of representations into smaller spaces which are easier to handle. The following definition is from [CK7]. Definition 2. Let C be a GL(d)-invariant, irreducible, closed subvariety of rep(a, d), and assume C is -semistable. Consider a collection (C i rep(a, d i )) i of -stable irreducible components such that C i C j for i j, along with a collection of multiplicities (m i Z >0 ) i, and set C = C m Cr mr. We say that (C i, m i ) i is a -stable decomposition of C if, for a general representation M C ss, its corresponding -polystable representation M is in C, and write (2) C = m C... m r C r. Any GL(d)-invariant, irreducible, closed subvariety C of rep(a, d) with C ss admits a -stable decomposition [CK7, Proposition 3]. This decomposition controls the geometry of irreducible components of moduli spaces in the following sense. Theorem 3. [CK7, Theorem ] Let A be a finite-dimensional algebra and let C rep(a, d) ss be a GL(d)-invariant, irreducible, closed subvariety. Let C = m C... m r C r be a -stable decomposition of C where C i rep(a, d i ), i r, are pairwise distinct -stable irreducible components, and define C = C m Cr mr. (a) If M(C) ss is an irreducible component of M(A, d)ss, then ss M( C) = M(C) ss. 4

5 (b) If C is an orbit closure, then M(C m Cr mr ) ss M(C m 2 2 Cr mr ) ss. (c) Assume now that none of the C i are orbit closures. Then there is a natural morphism Ψ: S m (M(C ) ss )... S mr (M(C r ) ss ) M( ss which is finite, and birational. In particular, if M( C) is normal then Ψ is an isomorphism. Note that given any (non-empty) moduli space M(A, d) ss are all of the form M(C) ss C) ss, its irreducible components with C a -semistable irreducible component of rep(a, d). and not just those Thus, the theorem covers all the irreducible components of M(A, d) ss of some special form. We also remark that for a Schur-tame algebra, each M(C i ) ss appearing in the theorem has dimension 0 if C i is an orbit closure, and dimension otherwise. Therefore, the dimension of M(C) ss not orbit closures. is precisely the sum of the multiplicities of the components which are 3. VARIETIES OF CIRCULAR COMPLEXES 3.. Definition. Fix a positive integer l and an l-tuple of positive integers n = (n i ) (for convenience in indices, we denote the residue class of an integer i modulo lz by the same letter l). We are interested in the variety Comp(n) := {(A i ) Mat ni+ n i (K) A i+ A i = 0, i Z/lZ}, called the variety of circular complexes associated to n. By convention, if l =, we get the variety of matrices A of size n 0 n 0 with A 2 = 0. Our goal in this section is to describe the irreducible components of Comp(n). First, we explain how to label the irreducible components of Comp(n) by maximal rank sequences. For this initial step, we follow the same strategy as that for dealing with varieties of (non-circular) complexes (see for example the presentation in [CW3, Section 3]). It is useful to view Comp(n) as a representation variety for the following bound quiver. Consider the oriented cycle C with vertex set Z/lZ: C : l a l 0 a 0 together with the admissible set of relations R := {a i+ a i i Z/lZ}. Viewing n as a dimension vector of C, Comp(n) is precisely the representation variety rep(kc/ R, n). Furthermore, KC/ R is a representation-finite algebra whose indecomposable representations are: 5

6 () the simples S i, i Z/lZ; (2) for each i Z/lZ, the representation E i,i+ defined to be K at vertices i, i +, the identity map along the arrow a i, and zero at all the other vertices and arrows. By convention, in case l =, C is just the one-loop quiver with R = {a 2 } where a denotes the loop of C. The indecomposable representations in this case are the simple S 0 at vertex 0 of C and the 2-dimensional representation J 2,0, given by the 2 2 nilpotent Jordan block along the arrow a. Consequently, if l >, any n-dimensional representation of (C, R), M can be written as: M E t i i,i+ where the non-negative integers t i and s i, i Z/lZ, satisfy the following conditions: S s i i, t i + t i + s i = n i and t i = rank M(a i ), i Z/lZ. If l =, these equations become 2t 0 + s 0 = n 0 and t 0 = rank M(a) where M J t 0 2,0 S s 0 0. In either case, we can see that M is uniquely determined, up to isomorphism, by its dimension vector and the rank sequence (rank M(a i )). In what follows, by a rank sequence for n, we mean a sequence r = (r i ) such that there exists an M rep(kc/ R, n) with r i = rank M(a i ), i Z/lZ; in particular, such an r must satisfy r i + r i n i for all i Z/lZ Properties of irreducible components. Now, for a rank sequence r for n, consider the closed subvariety: Comp(n, r) := {(A i ) Comp(n) rank A i r i, i Z/lZ}. From the discussion above, we get that Comp 0 (n, r) := {(A i ) Comp(n) rank A i = r i, i Z/lZ} is the GL(n)-orbit in Comp(n) of M 0 (n, r) := E r i i,i+ S n i r i r i i. A priori, the rank conditions defining Comp(n, r) could lead to a variety with more than one irreducible component; we will now show that this does not happen assuming the rank conditions are maximal. Lemma 4. The irreducible components of Comp(n) are precisely those Comp(n, r) with r a maximal (with respect to the coordinate-wise order) rank sequence for n. Moreover, they are normal varieties. Proof. Since every irreducible component must have a dense orbit, it is enough to show that Comp(n, r) = Comp 0 (n, r) for all dimension vectors n and rank sequences r. The containment is immediate from semi-continuity of rank; to show the opposite containment, we take an arbitrary point of Comp(n, r) and produce an explicit degeneration from Comp 0 (n, r) to that point. Indeed, 6

7 let M Comp(n, r), and set r i := rank M(a i ) and ɛ i := r i r i for all i Z/lZ. Then M belongs to the GL(n)-orbit of M 0 (n, r ) := E r i i,i+ Next, for each λ K, consider the representation E r i i,i+ S n i r i r i i S n i r i r i i. E i,i+ (λ) ɛ i, where E i,i+ is K at vertices i and i +, λ along the arrow a i, and zero elsewhere. This representation is isomorphic to M 0 (n, r) for λ 0, and to M 0 (n, r ) when λ = 0. So, we get that M GL(n)M 0 (n, r) = Comp 0 (n, r). This proves our claim that Comp(n, r) = Comp 0 (n, r). In particular, Comp(n, r) is an irreducible closed subvariety of Comp(n) for any rank sequence r for n. To see normality, every irreducible component of Comp(n) is an orbit closure since KC/ R is representation-finite. Orbit closures in varieties of circular complexes are examples of orbit closures of nilpotent representations of cyclicly-oriented type à quivers, so [Lus90, Theorem.3] gives that irreducible components of Comp(n) are locally isomorphic to an affine Schubert variety of type A. These varieties are known to be normal, for example by [Fal03, Theorem 8]. We need a dimension bound for irreducible components of varieties of circular complexes. Lemma 5. The dimension of Comp(n, r) is less than or equal to l 2 i=0 n2 i. Proof. Consider the product of flag varieties F l := F lag(r i, n i r i, V i ) where F lag(r i, n i r i, V i ) denotes a two step flag variety (which becomes a Grassmannian if r i + r i = n i ) for each i Z/lZ. Now consider the incidence variety: Z(n, r) := {(A i, (R i, R 2 i )) Comp(n, r) F l Im(A i ) R i R 2 i Ker(A i ), i Z/lZ}. We have the two projections: Z(n, r) p q F l Comp(n, r) The projection p makes Z(n, r) a vector bundle over F lag(r i, n i r i, V i ), so Z(n, r) is nonsingular, and the map q is a birational isomorphism, since it is an isomorphism over Comp 0 (n, r). In particular, dim Comp(n, r) = dim Z(n, r) = dim F l + dim p ((R i, R 2 i ) ) 7

8 where (Ri, Ri 2 ) is an arbitrary flag in F l. For such a fixed flag, p ((Ri, Ri 2 ) is isomorphic to Hom K (V i /Ri 2, Ri+), which has dimension ri 2. Meanwhile, the formula for the dimension of a flag variety (see for example [Bri05,.2]) in this case gives dim F l = (r i + r i )(n i r i r i ) + r i r i. Therefore, dim Comp(n, r) = (r i + r i )(n i r i r i ) + r i r i + ri 2 = (r i + r i )(n i r i ). Let k i = n i r i r i. Note that with this notation, n 2 i = (r i + r i ) 2 + 2k i (r i + r i ) + k 2 i, while dim Comp(n, r) = (r i + r i )(r i + k i ). Thus, we compute (suppressing the index of summation where convenient): n 2 i 2 dim Comp(n, r) = (r i + r i ) 2 + 2k i (r i + r i ) + k 2 i 2(r i + r i )(r i + k i ) = (r i + r i )(r i + r i + 2k i 2r i 2k i ) + k 2 i = (r i + r i )(r i r i ) + k 2 i = r 2 i r 2 i + k 2 i. Note that the first two sums are equal, since indices are taken modulo l, and the remaining sum is patently positive. Hence, the result follows. 4. REPRESENTATION VARIETIES OF SPECIAL BISERIAL ALGEBRAS AND PROOF OF THE MAIN RESULT Our main goal in this section is to check the normality condition in Theorem 3(c), when the algebra in question is special biserial. We do this by reducing the considerations to varieties of circular complexes, whose irreducible components we already know are normal varieties (see Section 3). 4.. Special biserial and complete gentle algebras. We begin by quickly recalling the definition of a special biserial bound quiver algebra (see [SW83]). A bound quiver (Q, R) is called a special biserial bound quiver if: (SB) for each vertex v Q 0 there are at most two arrows with head v, and at most two arrows with tail v; (SB2) for every arrow a Q, there exists at most arrow b Q such that ab / R, and there exists at most one arrow c Q such that ca / R. A special biserial bound quiver (Q, R) is called gentle if the following additional properties hold (see [ASS06, IX.6]): (SB3) if a and a 2 are two arrows with the same tail v then, for any arrow b with head v, precisely one of the a b and a 2 b belongs to R; 8

9 (SB4) if b and b 2 are two arrows with the same head v then, for any arrow a with tail v, precisely one of the ab and ab 2 belongs to R; (SB5) R consists of paths of length two. If (Q, R) is gentle, we call KQ/ R a gentle bound quiver algebra. Any finite-dimensional algebra isomorphic to a gentle bound quiver algebra is called a gentle algebra. An algebra obtained from a gentle algebra by adding only mononial relations is known as a string algebra, and by adding arbitrary relations is a special biserial algebra. The finitedimensional indecomposable representation for these algebras are well-known. Specifically, an indecomposable representation is either a projective, or string, or band representation (see [BR87], [Rin75]). As explained by Ringel in [Rin], a special biserial algebra can be viewed as a quotient of a rather special infinite-dimensional gentle algebra, called a complete gentle algebra. Definition 6. Let Q be a quiver and R a finite set of monomial relations of length two. We say that (Q, R ) is a complete gentle quiver with relations if for every vertex i Q 0, there are precisely two arrows ending at i and precisely two arrows starting at i, and for every arrow a Q, there is precisely one arrow a Q and precisely one arrow a Q such that aa and a a belong to R. A complete gentle algebra is an algebra isomorphic to KQ / R with (Q, R ) a complete gentle quiver with relations. Note that a complete gentle algebra is infinite-dimensional. To describe the finite-dimensional indecomposable representations of complete gentle algebras, one uses the recipe developed for dealing with finite-dimensional gentle/string algebras. In particular, the finite-dimensional indecomposable representations are given again by bands and strings. This is due to the work of Ringel in [Rin75], and of Crawley- Boevey in [Cra3] where the more general case of finitely controlled or pointwise artinian indecomposable representations over infinite-dimensional string algebras is discussed Representation varieties of complete gentle algebras. Let (Q, R ) be a complete gentle quiver with relations, Λ = KQ / R its complete gentle algebra, and d Z Q 0 0 a dimension vector. In what follows, by an effective oriented cycle of (Q, R ), we mean an oriented cycle C = a... a n of Q such that a i a j for i j, and a a 2,..., a n a n, a n a R. (If n =, we say that C = a is an effective oriented cycle if a 2 R ). Since each arrow belongs to a unique effective oriented cycle, Q can be written as a disjoint union of subsets of the form {a i } n i= (n varying with the subset) where C = a... a n is an effective oriented cycle. Therefore, the representation variety rep(λ, d) is a product of varieties of circular complexes. Hence, the irreducible components of rep(λ, d) are normal varieties by Lemma 4. This is, in fact, one of the key advantages of working with complete gentle algebras. To describe the irreducible components in more concrete terms, let us recall that a sequence r = (r a ) a Q of non-negative integers is called a rank sequence for d if there exists an M rep(λ, d) with r a = rank M(a), a Q. Note that this condition implies that r a + r b d(ta) for any two arrows a, b with ab R. A rank sequence for d which is maximal with respect to the coordinate-wise order is called a maximal rank sequence for d. It follows from Section 3 that for any rank sequence r for d, the set rep(λ, d, r) := {M rep(λ, d) rank M(a) r a, a Q } 9

10 is an irreducible closed subvariety of rep(λ, d). Moreover, by Lemma 4, the irreducible components of rep(λ, d) are precisely those rep(λ, d, r) with r a maximal rank sequence for d. Lemma 7. Let Λ be a complete gentle algebra and r a rank sequence for a dimension vector d. Then dim rep(λ, d, r) i Q 0 d(i) 2 = dim GL(d). Proof. We have noted above that rep(λ, d, r) is isomorphic to a product of varieties of complexes, say j Comp(nj, r j ). Then we have by Lemma 5 that dim rep(λ, d, r) = j dim Comp(n j, r j ) j 2 (n j i )2. Now for each vertex i Q 0, the value d(i) appears exactly twice among the values (n j i ), since each vertex of Q 0 is a vertex for precisely two varieties of complexes (or the same one twice). So the last double sum simplifies to i Q 0 d(i) 2. We also need the concept of a regular irreducible component (for Λ or any special biserial algebra). By this, we mean an irreducible component C that contains a regular representation (i.e. a representation that breaks into a direct sum of only band representations). We have the following very useful result, which is used in the proof of Proposition 9. Lemma 8. Let Λ be a complete gentle algebra, d a dimension vector, and r a rank sequence. Assume that rep(λ, d, r) is a regular irreducible component of rep(λ, d). Then: r a + r b = d(ta), a, b Q with ab R. Proof. Choose a regular representation M rep(λ, d, r). We claim that rank M(a) = r a, a Q. Indeed, recall that for any finite-dimensional representation X: rank X(a) = dim K X s, a Q where s is the number of strings occurring in a direct sum decomposition of X into indecomposables. (This follows immediately from the construction of the string and band representations.) Consequently, if X rep(λ, d, r) is so that rank X(a) = r a for all a Q, then i Q 0 d(i) = a Q rank M(a) a Q and so rank M(a) = rank X(a) = r a for all a Q. Next, given an effective oriented cycle C, we have: a C rank M(a) 2 rank X(a) i Q 0 i C 0 d(i) + j C 0 d(j), i d(i), where C is the set of arrows of C, C 0 is the set of vertices of C where C does not cross itself, and C 0 is the set of vertices of C where C crosses itself twice. The equality holds if and only if rank M(a) + rank M(b) = d(ta) for any two arrows a and b of C with ab R. 0

11 So, we get that: i Q 0 d(i) = a Q rank M(a) = C rank M(a) d(i) + d(j) = d(i), 2 a C C i C 0 j C 0 i Q 0 where last two sums are over all effective oriented cycles in (Q, R ). Hence, we must have r a + r b = d(ta) for all arrows a and b of Q with ab R Proof of the main result. For two given irreducible components C rep(a, f) and C rep(a, f ), we set: We are now ready to prove: hom A (C, C ) = min{dim K Hom A (X, Y ) (X, Y ) C C }. Proposition 9. Let A = KQ/I be an arbitrary special biserial bound quiver algebra. Let C i rep(a, d i ), i m, be irreducible components such that a general representation in each C i is Schur, and that hom A (C i, C j ) = 0 for all i, j m. Then C := C... C m is a normal variety. Proof. Take a complete gentle quiver with relations (Q, R ) such that Q 0 = Q 0 and A is a quotient of Λ := KQ / R by an ideal generated by arrows and admissible relations. We will find a maximal rank sequence r for a dimension vector d for Λ such that C = rep(λ, d, r), with the latter normal by Lemma 4. Now, for each i m, we have that dim C i = dim GL(d i ) by the same arguments in [CC5b, Lemma 3], since C i is not an orbit closure. But, we can also view C i rep(λ, d i ), where the maximal dimension of an irreducible component is dim GL(d i ) by Lemma 7. Therefore, C i has to be an irreducible component of rep(λ, d i ) for each i m; in particular, each C i is a normal variety. Next, for each i m, write C i = rep(λ, d i, r i ) where r i = (ra) i a Q is a maximal rank sequence for d i with ra i = 0, a Q \Q. For r := r +...+r m, we have that r a +r b = d(ta) for all arrows a, b with ab R by Lemma 8. Consequently, r is a maximal rank sequence for d, and so rep(λ, d, r) is an irreducible component of rep(λ, d). On the other hand, we have that C = m i= rep(λ, d i, r i ) rep(λ, d, r), with the latter having dimension at most dim GL(d) by Lemma 7. So we will show that dim C = dim GL(d) as well, forcing equality. Let M C be a general element, so that M is a direct sum of Schur representations with no nonzero morphisms between these summands. Thus dim K End A (M) = m = dim Stab GL(d) (M). We also know by the general relation between dimensions of orbits and stabilizers that (3) dim GL(d) = dim GL(d) M + dim Stab GL(d) (M) = dim GL(d) M + m. On the other hand, C has a dense m-parameter family of distinct orbits, so for a general M C we have that (4) dim C = dim GL(d) M + m. Combining equations (3) and (4) then finishes the proof.

12 Proof of Theorem. Let Y be an arbitrary irreducible component of M(A, d) ss. Then there exists an irreducible component C of rep(a, d) such that Y = M(C) ss. Consider the - stable decomposition C = m C... m l C l as in Definition 2, so that by Theorem 3(a)(c) we have a morphism Ψ: S m (M(C ) ss )... S mr (M(C r ) ss ) M(C) ss which is surjective, finite, and birational. By Theorem 3(a) we can assume that C = C m... C m l l and no C i is an orbit closure, so each C i must contain a dense family of band representations. Next, we claim that hom A (C i, C j ) = 0 for all i, j l. Indeed, for any i, j l, simply choose two non-isomorphic -stable representations X i and Y j from C i and C j, respectively; this is always possible since each C i is not an orbit closure. Then Hom A (X i, Y j ) = 0 and so hom A (C i, C j ) = 0. It now follows from Proposition 9 that C (keeping in mind the reductions above) is normal. Since A is tame, we already know that each M(C i ) ss is a rational projective curve (see, for example, [CC5a, Proposition 2]). But M(C i ) ss is also normal since C i is normal by the m = case in Proposition 9; hence M(C i ) ss P for all i l. We conclude that M(C) ss l i= Pm i. REFERENCES [ABCJP0] I. Assem, T. Brüstle, G. Charbonneau-Jodoin, and P.-G. Plamondon, Gentle algebras arising from surface triangulations, Algebra Number Theory 4 (200), no. 2, MR [ASS06] I. Assem, D. Simson, and A. Skowroński, Elements of the representation theory of associative algebras. Vol. : Techniques of representation theory, London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge, MR MR (2006j:6020) [BC09] F. M. Bleher and T. Chinburg, Pullback moduli spaces, Comm. Algebra 37 (2009), no. 4, MR [BCHZ5] F. M. Bleher, T. Chinburg, and B. Huisgen-Zimmermann, The geometry of finite dimensional algebras with vanishing radical square, J. Algebra 425 (205), MR [Bob08] G. Bobiński, On the zero set of semi-invariants for regular modules over tame canonical algebras, J. Pure Appl. Algebra 22 (2008), no. 6, MR MR [Bob2] G. Bobinski, Semi-invariants for concealed-canonical algebras, Preprint available at arxiv: [math.rt], 202. [Bob4] G. Bobiński, On moduli spaces for quasitilted algebras, Algebra Number Theory 8 (204), no. 6, MR [Bon98] Klaus Bongartz, Some geometric aspects of representation theory, Algebras and modules, I (Trondheim, 996), CMS Conf. Proc., vol. 23, Amer. Math. Soc., Providence, RI, 998, pp. 27. MR (99j:6005) [BR87] M. C. R. Butler and C. M. Ringel, Auslander-Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra 5 (987), no. -2, MR MR (88a:6055) [Bri05] Michel Brion, Lectures on the geometry of flag varieties, Topics in cohomological studies of algebraic varieties, Trends Math., Birkhäuser, Basel, 2005, pp MR [BS99] G. Bobiński and A. Skowroński, Geometry of modules over tame quasi-tilted algebras, Colloq. Math. 79 (999), no., MR MR678 (2000i:4067) [CBS02] W. Crawley-Boevey and J. Schröer, Irreducible components of varieties of modules, J. Reine Angew. Math. 553 (2002), MR MR94482 (2004a:6020) [CC5a] A. T. Carroll and C. Chindris, Moduli spaces of modules of Schur-tame algebras, Algebr. Represent. Theory 8 (205), no. 4, MR

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14 [Zwa] Grzegorz Zwara, Singularities of orbit closures in module varieties, Representations of algebras and related topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 20, pp MR DEPAUL UNIVERSITY, DEPARTMENT OF MATHEMATICAL SCIENCES, CHICAGO, IL, USA address, Andrew T. Carroll: UNIVERSITY OF MISSOURI-COLUMBIA, MATHEMATICS DEPARTMENT, COLUMBIA, MO, USA address, Calin Chindris: UNIVERSITY OF IOWA, DEPARTMENT OF MATHEMATICS, IOWA CITY, IA, USA address, Ryan Kinser: UNIVERSITY OF CONNECTICUT, DEPARTMENT OF MATHEMATICS, STORRS, CT, USA address, Jerzy Weyman: 4