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Alan Muniz

Publications and source records attributed to Alan Muniz.

14 recordsLinked to original sources
arXiv2026

Minimal-Degree Foliations on Cominuscule Grassmannians

Given $X$ a cominuscule Grassmannian (or irreducible Hermitian symmetric space) and an integer $p,$ we compute the minimum $l(p)$ such that $H^0 (\Omega^p_X (l(p)))$ is not 0. This allows us to conclude that any codimension-one foliation of degree zero on a cominuscule Grassmannian is a pencil of hyperplanes, improving a result of the first and third authors with D. Faenzi. We also deduce the structure of codimension-one foliations of degree one. Finally, we provide families of examples of high codimensional foliations of minimal degree on classical Grassmannians, Lagrangian Grassmannians, Spinor varieties, and the Cayley plane.

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arXiv2026

Remarks on the second Chern class of a foliation

We bound the second Chern class of the tangent sheaf of a codimension-one foliation. Equivalently, we bound the degree of the pure codimension-two part of the singular scheme. In particular, for a degree-$d$ foliation on the projective space, the codimension-two part of its singular scheme must have degree at least $d+1$. Moreover, equality holds only for rational foliations of type $(1,d+1)$. These bounds involve counting an invariant related to first-order unfoldings of 2-dimensional foliated singularities.

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arXiv2025

Degree-one foliations on complete intersections

We prove that, under mild restrictions, the space of codimension-one foliations of degree one on a smooth projective complete intersection has two irreducible components of logarithmic type. We also prove that the same conclusion holds for any smooth hypersurface of dimension at least three that is not a quadric threefold. The proof of these results follows essentially from a more general structure theorem for foliations on manifolds covered by lines.

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arXiv2023

On the minimal algebraic complexity of the rank-one approximation problem for general inner products

We study the algebraic complexity of Euclidean distance minimization from a generic tensor to a variety of rank-one tensors. The Euclidean Distance (ED) degree of the Segre-Veronese variety counts the number of complex critical points of this optimization problem. We regard this invariant as a function of inner products. We prove that Frobenius inner product is a local minimum of the ED degree, and conjecture that it is a global minimum. We prove our conjecture in the case of matrices and symmetric binary and $3\times 3\times 3$ tensors. We discuss the above optimization problem for other algebraic varieties, classifying all possible values of the ED degree. Our approach combines tools from Singularity Theory, Morse Theory, and Algebraic Geometry.

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arXiv2023

Codimension one foliations on homogeneous varieties

The aim of this paper is to study codimension one foliations on rational homogeneous spaces, with a focus on the moduli space of foliations of low degree on Grassmannians and cominuscule spaces. Using equivariant techniques, we show that codimension one degree zero foliations on (ordinary, orthogonal, symplectic) Grassmannians of lines, some spinor varieties, some Lagrangian Grassmannians, the Cayley plane (an $E_6$-variety) and the Freudenthal variety (an $E_7$-variety) are identified with restrictions of foliations on the ambient projective space. We also provide some evidence that such results can be extended beyond these cases.

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arXiv2022

P-Forms from Syzygies

These notes aim to develop a tool for constructing polynomial differential $p$-forms vanishing on prescribed loci through syzygies of homogeneous ideals. Examples are provided through implementing this method in Macaulay2, particularly examples of instanton bundles of charges $4$ and $5$ on $\mathbb{P}^3$ that arise in this construction.

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arXiv2022

Holomorphic foliations of degree two and arbitrary dimension

We prove a complete classification of degree-$2$ foliations on $\mathbb{P}^n$ in any dimension, assuming they are not algebraically integrable. If $\mathcal{F}$ is such a foliation, then either $\mathcal{F}$ is the linear pull-back of a degree-$2$ foliation by curves on $\mathbb{P}^{n-k+1}$, or a logarithmic foliation of type $(1^{n-k+1},2)$, or a logarithmic foliation of type $(1^{n-k+3})$, or the linear pull-back of a degree-$2$ foliation of dimension $2$ on $\mathbb{P}^{n-k+2}$ tangent to an action of the Lie algebra $\mathfrak{aff}(\mathbb{C})$. Meanwhile, we prove that any $2$-dimensional foliation tangent to a global vector field must satisfy that its tangent sheaf is either not locally free or has a direct summand isomorphic to $\mathcal{O}_{\mathbb{P}^{n}}(a)$, with $a\in\{0,1\}$. As a byproduct of our classification, we describe the geometry of Poisson structures on $\mathbb{P}^{4}$ with generic rank two.

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arXiv2022

On the moduli of logarithmic connections on elliptic curves

We describe moduli spaces of logarithmic rank $2$ connections on elliptic curves with $n \geq 1$ poles and generic residues. In particular, we generalize a previous work by the first and second named authors. Our main approach is to analyze the underlying parabolic bundles; their stability and instability play a major role.

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arXiv2022

Moduli of Distributions via Singular Schemes

Let $X$ be a smooth projective variety. We show that the map that sends a codimension one distribution on $X$ to its singular scheme is a morphism from the moduli space of distributions into a Hilbert scheme. We describe its fibers and, when $X = \mathbb{P}^n$, compute them via syzygies. As an application, we describe the moduli spaces of degree 1 distributions on $\mathbb{P}^3$. We also give the minimal graded free resolution for the ideal of the singular scheme of a generic distribution on $\mathbb{P}^3$.

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arXiv2021

Logarithmic sheaves of complete intersections

We define logarithmic tangent sheaves associated with complete intersections in connection with Jacobian syzygies and distributions. We analyse the notions of local freeness, freeness and stability of these sheaves. We carry out a complete study of logarithmic sheaves associated with pencils of quadrics and compute their projective dimension from the classical invariants such as the Segre symbol and new invariants (splitting type and degree vector) designed for the classification of irregular pencils. This leads to a complete classification of free (equivalently, locally free) pencils of quadrics. Finally we produce examples of locally free, non free pencils of surfaces in P3 of any degree k at least 3, answering (in the negative) a question of Calvo-Andrade, Cerveau, Giraldo and Lins Neto about codimension foliations on P3 .

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arXiv2020

Foliations on the projective plane with finite group of symmetries

Let $\mathcal{F}$ denote a singular holomorphic foliation on $\mathbb{P}^2$ having a finite automorphism group $\mbox{aut}(\mathcal{F})$. Fixed the degree of $\mathcal{F}$, we determine the maximal value that $|\mbox{aut}(\mathcal{F})|$ can take and explicitly exhibit all the foliations attaining this maximal value. Furthermore, we classify the foliations with large but finite automorphism group.

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arXiv2017

Polynomial bounds for automorphisms groups of foliations

Let $(X, \mathcal{F})$ be a foliated surface and $G$ a finite group of automorphisms of $X$ that preserves $\mathcal{F}$. We investigate invariant loci for $G$ and obtain upper bounds for its order that depends polynomially on the Chern numbers of $X$ and $\mathcal{F}$. As a consequence, we estimate the order of the automorphism group of some foliations under mild restrictions. We obtain an optimal bound for foliations on the projective plane which is attained by the automorphism groups of the Jouanolou's foliations.

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