Moduli Spaces of Modules of Schur-Tame Algebras

Abstract

ABSTRACT. In this paper, we first show that for an acyclic gentle algebra A, the irreducible components of any moduli space of A-modules are products of projective spaces. Next, we show that the nice geometry of the moduli spaces of modules of an algebra does not imply the tameness of the representation type of the algebra in question. Finally, we place these results in the general context of moduli spaces of modules of Schur-tame algebras. More specifically, we show that for an arbitrary Schur-tame algebra A and θ-stable irreducible component C of a module variety of A-modules, the moduli space M(C) ss θ is either a point or a rational projective curve.

Representation Varieties of Algebras With Nodes

“…One motivation to study the normality of irreducible components of representation varieties comes from the main theorem of [CK18], which gives decompositions of moduli spaces of semistable representations. These decompositions were first proven for algebras of the form A = kQ with Q acyclic in [DW11] (see also [CB02]), then extended to certain classes of nonhereditary algebras in works such as [Chi13,CC15,CCKW20]. We discuss such applications and related semistability results in §5.…”

Decomposing moduli of representations of finite-dimensional algebras

“…The notion of θ-stable decomposition was introduced by Derksen and Weyman [DW11] for the case that A = KQ where Q is acyclic (so that all rep(A, d) are just vector spaces). An extension to GL(d)-invariant irreducible subvarieties C ⊆ rep(A, d) when A is an arbitrary algebra was given in [Chi11b,Chi13,CC15].…”

Moduli Spaces of Representations of Special Biserial Algebras

“…Recall that a Schur-tame algebra is an algebra such that, in each dimension vector, all Schur representations (except possibly finitely many) come in a finite number of 1parameter families (see [CC15a, Definition 3] for more details). For a Schur-tame algebra, each M(C i ) ss θ appearing in the theorem has dimension 0 if C i is an orbit closure, and dimension 1 otherwise (see [CC15a,Proposition 12]). Therefore, the dimension of M(C) ss θ is precisely the sum of the multiplicities of the components which are not orbit closures.…”