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Daniel Labardini-Fragoso

Publications and source records attributed to Daniel Labardini-Fragoso.

At least 19 recordsLinked to original sources
arXiv2023

Bangle functions are the generic basis for cluster algebras from punctured surfaces with boundary

We prove that for any possibly-punctured surface with non-empty boundary $\mathbf{\Sigma}=(\Sigma, \mathbb{M}, \mathbb{P})$, and any tagged triangulation $T$ of $\mathbf{\Sigma}$ in the sense of Fomin--Shapiro--Thurston, the coefficient-free bangle functions of Musiker--Schiffler--Williams coincide with the coefficient-free generic Caldero--Chapoton functions arising from the Jacobian algebra of the quiver with potential $(Q(T), W(T))$ associated to $T$ by Cerulli Irelli and the second author. When the set of boundary marked points $\mathbb{M}$ has at least two elements, Schr\"oer and the first two authors have shown, relying heavily on results of Mills, Muller and Qin, that the generic coefficient-free Caldero-Chapoton functions form a basis of the coefficient-free (upper) cluster algebra $\mathcal{A}(\mathbf{\Sigma})=\mathcal{U}(\mathbf{\Sigma})$. So, the set of bangle functions proposed by Musiker--Schiffler--Williams over ten years ago is indeed a basis.

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arXiv2023

Gentle algebras arising from surfaces with orbifold points, Part II: Locally free Caldero-Chapoton functions

We prove that in the skew-symmetrizable cluster algebras associated by Felikson-Shapiro-Tumarkin to unpunctured surfaces with orbifold points of order $2$ and a specific choice of weights, the Laurent expansion of any cluster variable with respect to any cluster coincides with the locally free Caldero-Chapoton function of a $\tau$-rigid representation of a gentle algebra. These cluster algebras are typically non-acyclic and of infinite type, whereas for polygons with one orbifold point one recovers cluster algebras of finite type $C$; so, our result is an ample extension of a seminal result established by Geiss-Leclerc-Schr\"oer for skew-symmetrizable cluster algebras of finite type and acyclic initial seeds. As the main means to achieve the result, we provide a generalization of Derksen-Weyman-Zelevinsky's mutation theory of loop-free quivers with potential to the quivers-with-loops with potential we associate to the triangulations of unpunctured surfaces with orbifold points, and study the relation with $\tau$-tilting theory. As a result of independent interest, we compute the aforementioned $\tau$-rigid representations explicitly. To this end, we show that the indecomposable $\tau$-rigid string modules arising from arcs on the surface, and the quasi-simple band modules arising from simple closed curves, are well-behaved under the mutations of representations we define in the paper, thus extending results of the first author's Ph.D. thesis.

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arXiv2023

Laminations of punctured surfaces as $\tau$-regular irreducible components

Let $\boldsymbol{\Sigma}:=(\Sigma,\mathbb{M},\mathbb{P})$ be a surface with marked points $\mathbb{M}\subset\partial\Sigma\neq\varnothing$ on the boundary, and punctures $\mathbb{P}\subset\Sigma\setminus\partial\Sigma$, and $T$ an arbitrary tagged triangulation of $\boldsymbol{\Sigma}$ in the sense of Fomin-Shapiro-Thurston. The Jacobian algebra $A(T):=\mathcal{P}(Q(T), W(T))$ corresponding to the non-degenerate potential $W(T)$ defined by Cerulli Irelli and the second author is tame, as shown by Schr\"{o}er and the first two authors. In this paper, we show that there is a natural isomorphism $\pi_T:\operatorname{Lam}(\boldsymbol{\Sigma})\rightarrow\operatorname{DecIrr}^\tau(A(T))$ of tame partial KRS-monoids that intertwines dual shear coordinates with respect to $T$, and generic $g$-vectors of irreducible components. Here, $\operatorname{Lam}(\boldsymbol{\Sigma})$ is the set of laminations of $\boldsymbol{\Sigma}$ considered by Musiker-Schiffler-Williams, with the disjoint union of non-intersecting laminations as partial monoid operation. On the other hand, $\operatorname{DecIrr}^\tau(A(T))$ denotes the set of generically $\tau$-regular irreducible components of the decorated representation varieties of $A(T)$, with the direct sum of generically $E$-orthogonal irreducible components as partial monoid operation, where $E$ is the symmetrized $E$-invariant of Derksen-Weyman-Zelevinsky, $E(-,\bullet)=\dim\operatorname{Hom}_{A(T)}(-,\tau(\bullet))+\dim\operatorname{Hom}_{A(T)}(\bullet,\tau(-))$.

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arXiv2023

On the resolution of kinks of curves on punctured surfaces

Let $(\Sigma,\mathbb{M},\mathbb{P})$ be a surface with marked points $\mathbb{M}\subseteq \partial\Sigma\neq\varnothing$ and punctures $\mathbb{P}\subseteq\Sigma\setminus\partial\Sigma$. In this paper we show that for every curve $\gamma$ on $\Sigma\setminus\mathbb{P}$, the curve obtained by resolving the kinks of $\gamma$ in any order is uniquely determined, up to homotopy in $\Sigma\setminus\mathbb{P}$, by the $2$-orbifold homotopy class of $\gamma$, in which the punctures are interpreted to be orbifold points of order $2$. Our proof resorts to an application of the Diamond Lemma.

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arXiv2023

Semilinear clannish algebras arising from surfaces with orbifold points

Semilinear clannish algebras have been recently introduced by the first author and Crawley-Boevey as a generalization of Crawley-Boevey's clannish algebras. In the present paper, we associate semilinear clannish algebras to the (colored) triangulations of a surface with marked points and orbifold points, and exhibit a Morita equivalence between these algebras and the Jacobian algebras constructed a few years ago by Geuenich and the second author.

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arXiv2023

Semicontinuous maps on module varieties

We study semicontinuous maps on varieties of modules over finite-dimensional algebras. We prove that truncated Euler maps are upper or lower semicontinuous. This implies that $g$-vectors and $E$-invariants of modules are upper semicontinuous. We also discuss inequalities of generic values of some upper semicontinuous maps.

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arXiv2023

Gentle algebras arising from surfaces with orbifold points of order 3, Part I: scattering diagrams

To each triangulation of any surface with marked points on the boundary and orbifold points of order three, we associate a quiver (with loops) with potential whose Jacobian algebra is finite dimensional and gentle. We study the stability scattering diagrams of such gentle algebras and use them to prove that the Caldero--Chapoton map defines a bijection between reachable $τ$-rigid pairs and cluster monomials of the generalized cluster algebra associated to the surface by Chekhov and Shapiro.

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arXiv2022

Landau-Ginzburg potentials via projective representations

We interpret the Landau-Ginzburg potentials associated to Gross-Hacking-Keel-Kontsevich's partial compactifications of cluster varieties as F-polynomials of projective representations of Jacobian algebras. Along the way, we show that both the projective and the injective representations of Jacobi-finite quivers with potential are well-behaved under Derksen-Weyman-Zelevinsky's mutations of representations.

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arXiv2021

Schemes of modules over gentle algebras and laminations of surfaces

We study the affine schemes of modules over gentle algebras. We describe the smooth points of these schemes, and we also analyze their irreducible components in detail. Several of our results generalize formerly known results, e.g. by dropping acyclicity, and by incorporating band modules. A special class of gentle algebras are Jacobian algebras arising from triangulations of unpunctured marked surfaces. For these we obtain a bijection between the set of generically tau-reduced decorated irreducible components and the set of laminations of the surface. As an application, we get that the set of bangle functions (defined by Musiker-Schiffler-Williams) in the upper cluster algebra associated with the surface coincides with the set of generic Caldero-Chapoton functions.

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arXiv2021

Derived categories of skew-gentle algebras and orbifolds

Skew-gentle algebras are a generalisation of the well-known class of gentle algebras with which they share many common properties. In this work, using non-commutative Gröbner basis theory, we show that these algebras are Koszul and that the Koszul dual is again skew-gentle. We give a geometric model of their bounded derived categories in terms of polygonal dissections of orbifold surfaces establishing a correspondence between curves in the orbifold and indecomposable objects. Moreover, we show that the orbifold dissections encode homological properties of skew-gentle algebras such as their singularity categories, their Gorenstein dimensions and derived invariants such as the determinant of their q-Cartan matrices.

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arXiv2020

Quivers with potentials associated to triangulations of closed surfaces with at most two punctures

We tackle the classification problem of non-degenerate potentials for quivers arising from triangulations of surfaces in the cases left open by Geiss-Labardini-Schröer. Namely, for once-punctured closed surfaces of positive genus, we show that the quiver of any triangulation admits infinitely many non-degenerate potentials that are pairwise not weakly right-equivalent; we do so by showing that the potentials obtained by adding the 3-cycles coming from triangles and a fixed power of the cycle surrounding the puncture are well behaved under flips and QP-mutations. For twice-punctured closed surfaces of positive genus, we prove that the quiver of any triangulation admits exactly one non-degenerate potential up to weak right-equivalence, thus confirming the veracity of a conjecture of the aforementioned authors.

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arXiv2020

Generic Caldero-Chapoton functions with coefficients and applications to surface cluster algebras

We realize Derksen-Weyman-Zelevinsky's mutations of representations as densely-defined regular maps on representation spaces, and study the generic values of Caldero-Chapoton functions with coefficients, giving, for instance, a sufficient combinatorial condition for their linear independence. For a quiver with potential $(Q,S)$, we show that if $k$ is a vertex not incident to any oriented 2-cycle, then every generically $τ$-reduced irreducible component $Z$ of any affine variety of (decorated) representations has a dense open subset $U$ on which Derksen-Weyman-Zelevinsky's mutation of representations $μ_k$ can be defined consistently as a regular map to an affine variety of (decorated) representations of the Jacobian algebra of the mutated QP $μ_k(Q,S)$. Our techniques involve only basic linear algebra and elementary algebraic geometry, and do not require to assume Jacobi-finiteness. Thus, the paper yields a new and more general proof of the mutation invariance of generic Caldero-Chapoton functions, generalizing and providing a new natural geometric perspective on results of Derksen-Weyman-Zelevinsky and Plamondon. (For Jacobi-finite non-degenerate quivers with potential, this invariance was shown by Plamondon using the machinery of Ginzburg dg-algebras and Hom-finite generalized cluster categories.) We apply our results, together with results of Mills, Muller and Qin, to prove that for any choice of geometric coefficient systems, not necessarily of full rank, the cluster algebra associated to a possibly punctured surface with at least two marked points on the boundary has the generic Caldero-Chapoton functions as a basis over the Laurent polynomial ring of coefficients.

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arXiv2019

On a family of Caldero-Chapoton algebras that have the Laurent phenomenon

We realize a family of generalized cluster algebras as Caldero-Chapoton algebras of quivers with relations. Each member of this family arises from an unpunctured polygon with one orbifold point of order 3, and is realized as a Caldero-Chapoton algebra of a quiver with relations naturally associated to any triangulation of the alluded polygon. The realization is done by defining for every arc $j$ on the polygon with orbifold point a representation $M(j)$ of the referred quiver with relations, and by proving that for every triangulation $τ$ and every arc $j\inτ$, the product of the Caldero-Chapoton functions of $M(j)$ and $M(j')$, where $j'$ is the arc that replaces $j$ when we flip $j$ in $τ$, equals the corresponding exchange polynomial of Chekhov-Shapiro in the generalized cluster algebra. Furthermore, we show that there is a bijection between the set of generalized cluster variables and the isomorphism classes of $E$-rigid indecomposable decorated representations of $Λ$.

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arXiv2017

Species with potential arising from surfaces with orbifold points of order 2, Part II: arbitrary weights

Let $\mathbfΣ=(Σ,M,O)$ be a surface with marked points and order-2 orbifold points which is either unpunctured or once-punctured closed, and $ω:O\rightarrow\{1,4\}$ a function. For each triangulation $τ$ of $\mathbfΣ$ we construct a cochain complex $C^\bullet(τ,ω)$. A colored triangulation is defined to be a pair consisting of a triangulation $τ$ and a 1-cocycle of $C^\bullet(τ,ω)$; the combinatorial notion of colored flip of colored triangulations is then defined as a refinement of the notion of flip of triangulations. Our main construction associates to each colored triangulation a species and a potential, and our main result shows that colored triangulations related by a colored flip have SPs related by the corresponding SP-mutation. We define the flip graph of $(Σ,M,O,ω)$, whose vertices are the pairs $(τ,x)$ with $τ$ a triangulation and $x$ a cohomology class in $H^1(C^\bullet(τ,ω))$, with an edge between $(τ,x)$ and $(σ,z)$ iff $(τ,ξ)$ and $(σ,ζ)$ are related by a colored flip for some cocycles $ξ$ and $ζ$ respectively representing $x$ and $z$. We prove that this graph is disconnected if $Σ$ is not contractible. For unpunctured surfaces we show that $(τ,ξ)$ and $(τ,ξ')$ yield isomorphic Jacobian algebras if and only if $[ξ]=[ξ']$ in cohomology. We prove that every SP-realization of any $(τ,ω)$ via a non-degenerate SP over a cyclic Galois extension with certain roots of unity is right-equivalent to one of the SPs we construct here. The species constructed here are species realizations of the $2^{|O|}$ skew-symmetrizable matrices assigned by Felikson-Shapiro-Tumarkin to any given $τ$. In the prequel to this paper we realized only one of these matrices via species, but therein we allowed the presence of arbitrarily many punctures.

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arXiv2016

Derived invariants for surface cut algebras II: the punctured case

For each algebra of global dimension 2 arising from the quiver with potential associated to a triangulation of an unpunctured surface, Amiot-Grimeland have defined an integer-valued function on the first singular homology group of the surface, and have proved that two such algebras of global dimension 2 are derived equivalent precisely when there exists an automorphism of the surface that makes their associated functions coincide. In the present paper we generalize the constructions and results of Amiot-Grimeland to the setting of arbitrarily punctured surfaces. As an application, we show that there always is a derived equivalence between any two algebras of global dimension 2 arising from the quivers with potential of (valency at least 2) triangulations of arbitrarily punctured polygons. While in the unpunctured case the quiver with potential of any triangulation admits cuts yielding algebras of global dimension at most 2, in the case of punctured surfaces the QPs of some triangulations do not admit cuts, and even when they do, the global dimension of the corresponding degree-0 algebra may exceed 2. In this paper we give a combinatorial characterization of each of these two situations.

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arXiv2016

Species with potential arising from surfaces with orbifold points of order 2, Part I: one choice of weights

We present a definition of mutations of species with potential that can be applied to the species realizations of any skew-symmetrizable matrix B over cyclic Galois extensions E/F whose base field F has a primitive [E:F]-th root of unity. After providing an example of a globally unfoldable skew-symmetrizable matrix whose species realizations do not admit non-degenerate potentials, we present a construction that associates a species with potential to each tagged triangulation of a surface with marked points and orbifold points of order 2. Then we prove that for any two tagged triangulations related by a flip, the associated species with potential are related by the corresponding mutation (up to a possible change of sign at a cycle), thus showing that these species with potential are non-degenerate. In the absence of orbifold points, the constructions and results specialize to previous work by the second author. The species constructed here for each triangulation $τ$ is a species realization of one of the several matrices that Felikson-Shapiro-Tumarkin have associated to $τ$, namely, the one that in their setting arises from choosing the number 1/2 for every orbifold point.

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