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Alfredo Nájera Chávez

Publications and source records attributed to Alfredo Nájera Chávez.

11 recordsLinked to original sources
arXiv2025

Deformation Theory for Finite Cluster Complexes

We study the deformation theory of the Stanley-Reisner rings associated to cluster complexes for skew-symmetrizable cluster algebras of geometric and finite cluster type. In particular, we show that in the skew-symmetric case, these cluster complexes are unobstructed, generalizing a result of Ilten and Christophersen in the $A_n$ case. We also study the connection between cluster algebras with universal coefficients and cluster complexes. We show that for a full rank positively graded cluster algebra $\mathcal{A}$ of geometric and finite cluster type, the cluster algebra $\mathcal{A}^{\mathrm{univ}}$ with universal coefficients may be recovered as the universal family over a partial closure of a torus orbit in a multigraded Hilbert scheme. Likewise, we show that under suitable hypotheses, the cluster algebra $\mathcal{A}^{\mathrm{univ}}$ may be recovered as the coordinate ring for a certain torus-invariant semiuniversal deformation of the Stanley-Reisner ring of the cluster complex. We apply these results to show that for any cluster algebra $\mathcal{A}$ of geometric and finite cluster type, $\mathcal{A}$ is Gorenstein, and $\mathcal{A}$ is unobstructed if it is skew-symmetric. Moreover, if $\mathcal{A}$ has enough frozen variables then it has no non-trivial torus-invariant deformations. We also study the Gröbner theory of the ideal of relations among cluster and frozen variables of $\mathcal{A}$. As a byproduct we generalize previous results in this setting obtained by Bossinger, Mohammadi and Nájera Chávez for Grassmannians of planes and $\text{Gr}(3,6)$.

math.AG↗
arXiv2024

Newton--Okounkov bodies and minimal models for cluster varieties

Let $Y$ be a (partial) minimal model of a scheme $V$ with a cluster structure. Under natural assumptions, for every choice of seed we associate a Newton--Okounkov body to every divisor on $Y$ supported on $Y \setminus V$ and show that these Newton--Okounkov bodies are positive sets in the sense of Gross, Hacking, Keel and Kontsevich \cite{GHKK}. This construction essentially reverses the procedure in loc. cit. that generalizes the polytope construction of a toric variety to the framework of cluster varieties. In a closely related setting, we consider cases where $Y$ is a projective variety whose universal torsor $\text{UT}_Y$ is a partial minimal model of a scheme with a cluster structure of type $\mathcal A$. If the theta functions parametrized by the integral points of the associated superpotential cone form a basis of the ring of algebraic functions on $\text{UT}_Y$ and the action of the torus $T_{\text{Pic}(Y)^*}$ on $\text{UT}_Y$ is compatible with the cluster structure, then for every choice of seed we associate a Newton--Okounkov body to every line bundle on $Y$. We prove that any such Newton--Okounkov body is a positive set and that $Y$ is a minimal model of a quotient of a cluster $\mathcal A$-variety by the action of a torus. Our constructions lead to the notion of the intrinsic Newton--Okounkov body associated to a boundary divisor in a partial minimal model of a scheme with a cluster structure. This provides a wide class of examples of Newton-Okoukov bodies exhibiting a wall-crossing phenomenon in the sense of Escobar--Harada \cite{EH20}. This approach includes the partial flag varieties that arise as minimal models of cluster varieties. For the case of Grassmannians, our approach recovers, up to interesting unimodular equivalences, the Newton--Okounkov bodies constructed by Rietsch--Williams in \cite{RW}.

math.AG↗
arXiv2024

Toric degenerations of cluster varieties and cluster duality

We introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: a cluster $\mathcal{X}$-variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed cluster $\mathcal{X}$-variety $\widehat{\mathcal{X}}$ to the toric variety associated to its $\mathbf{g}$-fan. Moreover, we show that the fibers of this family are stratified in a natural way, with strata the specially completed $\mathcal{X}$-varieties encoded by $\mathrm{Star}(τ)$ for each cone $τ$ of the $\mathbf{g}$-fan. These strata degenerate to the associated toric strata of the central fiber. We further show that the family is cluster dual to $\mathcal{A}_{\mathrm{prin}}$ of Gross-Hacking-Keel-Kontsevich, and the fibers cluster dual to $\mathcal{A}_t$. Finally, we give two applications. First, we use our construction to identify the Rietsch-Williams toric degeneration of Grassmannians with the Gross-Hacking-Keel-Kontsevich degeneration in the case of $\mathrm{Gr}_2(\mathbb{C}^5)$. Next, we use it to link cluster duality to Batyrev-Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.

math.AG↗
arXiv2024

The cluster complex for cluster Poisson varieties and representations of acyclic quivers

Let $\mathcal{X}$ be a skew-symmetrizable cluster Poisson variety. The cluster complex $Δ^+(\mathcal{X})$ was introduced by Gross, Hacking, Keel and Kontsevich. It codifies the theta functions on $\mathcal{X}$ that restrict to a character of a seed torus. Every seed ${ \bf s}$ for $\mathcal{X}$ determines a fan realization $Δ^+_{\bf s}(\mathcal{X})$ of $Δ^+(\mathcal{X})$. For every ${\bf s}$ we provide a simple and explicit description of the cones of $Δ^+_{\bf s}(\mathcal{X})$ and their facets using ${\bf c}$-vectors. Moreover, we give formulas for the theta functions parametrized by the integral points of $Δ^+_{ \bf s}(\mathcal{X})$ in terms of $F$-polynomials. In case $\mathcal{X}$ is skew-symmetric and the quiver $Q$ associated to ${\bf s}$ is acyclic, we describe the normal vectors of the supporting hyperplanes of the cones of $Δ^+_{\bf s}(\mathcal{X})$ using ${\bf g}$-vectors of (non-necessarily rigid) objects in $\mathsf{K}^{\rm b}(\text{proj} \; kQ)$.

math.RT↗
arXiv2021

Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras

Let $V$ be the weighted projective variety defined by a weighted homogeneous ideal $J$ and $C$ a maximal cone in the Gröbner fan of $J$ with $m$ rays. We construct a flat family over $\mathbb A^m$ that assembles the Gröbner degenerations of $V$ associated with all faces of $C$. This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated to a weight. We explain how our family can be constructed from Kaveh-Manon's recent work on the classification of toric flat families over toric varieties: it is the pull-back of a toric family defined by a Rees algebra with base $X_C$ (the toric variety associated to $C$) along the universal torsor $\mathbb A^m \to X_C$. We apply this construction to the Grassmannians ${\rm Gr}(2,\mathbb C^n)$ with their Plücker embeddings and the Grassmannian ${\rm Gr}\big(3,\mathbb C^6\big)$ with its cluster embedding. In each case, there exists a unique maximal Gröbner cone whose associated initial ideal is the Stanley-Reisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated to this cone. Further, for ${\rm Gr}(2,\mathbb C^n)$ we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as tropicalized cluster mutation.

math.AG↗
arXiv2021

Compactifications of cluster varieties and convexity

In [GHKK18], Gross-Hacking-Keel-Kontsevich discuss compactifications of cluster varieties from "positive subsets" in the real tropicalization of the mirror. To be more precise, let $\mathfrak{D}$ be the scattering diagram of a cluster variety $V$ (of either type -- $\mathcal{A}$ or $\mathcal{X}$), and let $S$ be a closed subset of $\left(V^\vee\right)^{\text{trop}}(\mathbb{R})$ -- the ambient space of $\mathfrak{D}$. The set $S$ is positive if the theta functions corresponding to the integral points of $S$ and its $\mathbb{N}$-dilations define an $\mathbb{N}$-graded subalgebra of $Γ(V, \mathcal{O}_V)[x]$. In particular, a positive set $S$ defines a compactification of $V$ through a Proj construction applied to the corresponding $\mathbb{N}$-graded algebra. In this paper we give a natural convexity notion for subsets of $\mathfrak{D}$, called "broken line convexity", and show that a set is positive if and only if it is broken line convex. The combinatorial criterion of broken line convexity provides a tractable way to construct positive subsets of $\mathfrak{D}$, or to check positivity of a given subset.

math.AG↗
arXiv2017

A 2-Calabi-Yau realization of finite-type cluster algebras with universal coefficients

We categorify various finite-type cluster algebras with coefficients using completed orbit categories associated to Frobenius categories. Namely, the Frobenius categories we consider are the categories of finitely generated Gorenstein projective modules over the singular Nakajima category associated to a Dynkin diagram and their standard Frobenius quotients. In particular, we are able to categorify all finite-type skew-symmetric cluster algebras with universal coefficients and finite-type Grassmannian cluster algebras. Along the way, we classify the standard Frobenius models of a certain family of triangulated orbit categories which include all finite-type $n$-cluster categories, for all integers $n\geq 1$.

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arXiv2015

On Frobenius (completed) orbit categories

Let ${\mathcal E}$ be a Frobenius category, ${\mathcal P}$ its subcategory of projective objects and $F:{\mathcal E} \to {\mathcal E}$ an exact automorphism. We prove that there is a fully faithful functor from the orbit category ${\mathcal E}/F$ into $\operatorname{gpr}({\mathcal P}/F)$, the category of finitely-generated Gorenstein-projective modules over ${\mathcal P}/F$. We give sufficient conditions to ensure that the essential image of ${\mathcal E}/F$ is an extension-closed subcategory of $\operatorname{gpr}({\mathcal P}/F)$. If ${\mathcal E}$ is in addition Krull-Schmidt, we give sufficient conditions to ensure that the completed orbit category ${\mathcal E} \ \widehat{\!\! /} F$ is a Krull-Schmidt Frobenius category. Finally, we apply our results on completed orbit categories to the context of Nakajima categories associated to Dynkin quivers and sketch applications to cluster algebras.

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arXiv2012

On the c-vectors of an acyclic cluster algebra

We prove that the set of c-vectors of the cluster algebra associated to an acyclic quiver Q coincides with the set of real Schur roots and their opposites in the root system associated to Q.

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arXiv2012

C-vectors and dimension vectors for cluster-finite quivers

Let $(Q,W)$ be a quiver with a non degenerate potential. We give a new description of the \textbf{c}-vectors of $Q$. We use it to show that, if $Q$ is mutation equivalent to a Dynkin quiver, then the set of positive $\mathbf{c}$-vectors of the cluster algebra associated to $Q^{\text{op}}$ coincides with the set of dimension vectors of the indecomposable modules over the Jacobian algebra of $(Q,W)$.

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