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Thiago Fassarella

Publications and source records attributed to Thiago Fassarella.

18 recordsLinked to original sources
arXiv2026

Foliations with small singular set in arbitrary characteristic

This paper investigates the geometry of foliations on smooth algebraic varieties over an algebraically closed field of arbitrary characteristic $p \ge 0$. We address several specific features of foliations in positive characteristic, aiming to highlight both similarities and differences with the characteristic zero case. First, we provide a complete classification of regular foliations on minimal rational and del Pezzo surfaces, establishing that in positive characteristic, such foliations are $p$-closed. However, there are regular foliations on weak del Pezzo surfaces, in characteristic 2, which are not $p$-closed. We extend the Camacho-Sad index and its associated sum formula to arbitrary characteristic, using it to study the behavior of foliations and distributions with small singular set. For foliations on projective spaces, we prove that the existence of an invariant hypersurface with a sufficiently small singular set forces the foliation to be $p$-closed and imposes strict divisibility conditions on the degree of its normal bundle. Finally, we establish versions of the Bott vanishing theorem in both Hodge cohomology and the Chow ring for $p$-closed foliations, providing a positive characteristic analogue to classical vanishing results in complex geometry.

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arXiv2026

Moduli of Higgs bundles over the two punctured elliptic curve

We study moduli spaces of Higgs bundles with two poles on an elliptic curve. We describe all singular fibers of the Hitchin map, including the nilpotent cone. To achieve this, we consider a modular map that lifts Higgs bundles with five poles on the Riemann sphere to Higgs bundles on the elliptic curve. This map is a two-sheeted covering and we analyze its Galois involution. We prove that the modular map is surjective and determine its ramification locus. In particular, we also obtain an explicit description of the singular locus of the moduli space.

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arXiv2025

Hyperplane arrangements and the Gauss map of a pencil

We show that the coefficients of the characteristic polynomial of a central hyperplane arrangement $\mathcal A$, coincide with the multidegrees of the Gauss map of a pencil of hypersurfaces naturally associated to $\mathcal A$. As a consequence, we obtain a proof of the Heron-Rota-Welsh conjecture for matroids representable over a field of characteristic zero.

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arXiv2023

Toric polar maps and characteristic classes

Given a hypersurface in a complex projective space, we prove that the multidegrees of its toric polar map agree, up to sign, with the coefficients of the Chern-Schwartz-MacPherson class of a distinguished open set, namely the complement of the union of the hypersurface and the coordinate hyperplanes. In particular, the degree of the toric polar map is given by the signed topological Euler characteristic of the distinguished open set. For plane curves, a precise formula for the degree of the toric polar map is obtained in terms of local invariants. Finally, we construct families, in arbitrary dimension, of irreducible hypersurfaces whose toric polar map is birational.

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arXiv2023

Hitchin fibration under ramified coverings

We are interested in studying the variation of the Hitchin fibration in moduli spaces of parabolic Higgs bundles, under the action of a ramified covering. Given a degree two map $π$ : Y $\rightarrow$ X between compact Riemann surfaces, we may pull back a Higgs bundle from X to Y , the lifted Higgs bundle tends to have many apparent singularities, then we perform a suitable birational transformation in order to eliminate them. This correspondence preserves the Hitchin fibrations and then its restriction to a general fiber gives a map between Abelian varieties. The aim of this paper is to describe this map.

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arXiv2023

Moduli of Higgs bundles over the five punctured sphere

We look at rank two parabolic Higgs bundles over the projective line minus five points which are semistable with respect to a weight vector $μ\in[0,1]^5$. The moduli space corresponding to the central weight $μ_c=(\frac{1}{2}, \dots, \frac{1}{2})$ is studied in details and all singular fibers of the Hitchin map are described, including the nilpotent cone. After giving a description of fixed points of the $\mathbb C^*$-action we obtain a proof of Simpson's foliation conjecture in this case. For each $n\ge 5$, we remark that there is a weight vector so that the foliation conjecture in the moduli space of rank two logarithmic connections over the projective line minus $n$ points is false.

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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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arXiv2020

Developable cubics in $\mathbb P^4$ and the Lefschetz locus in ${\rm GOR}(1,5,5,1)$

We provide a classification of developable cubic hypersurfaces in $\mathbb P^4$. Using the correspondence between forms of degree $3$ on $\mathbb P^4$ and Artinian Gorenstein $\mathbb K$-algebras, given by Macaulay-Matlis duality, we describe the locus in ${\rm GOR}(1,5,5,1)$ corresponding to those algebras which satisfy the Strong Lefschetz property.

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arXiv2020

A Torelli theorem for moduli spaces of parabolic vector bundles over an elliptic curve

Let $C$ be an elliptic curve, $w\in C$, and let $S\subset C$ be a finite subset of cardinality at least $3$. We prove a Torelli type theorem for the moduli space of rank two parabolic vector bundles with determinant line bundle $\mathcal O_C(w)$ over $(C,S)$ which are semistable with respect to a weight vector $\big(\frac{1}{2}, \dots, \frac{1}{2}\big)$.

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arXiv2019

On automorphisms of moduli spaces of parabolic vector bundles

Fix $n\geq 5$ general points $p_1, \dots, p_n\in\mathbb{P}^1$, and a weight vector $\mathcal{A} = (a_{1}, \dots, a_{n})$ of real numbers $0 \leq a_{i} \leq 1$. Consider the moduli space $\mathcal{M}_{\mathcal{A}}$ parametrizing rank two parabolic vector bundles with trivial determinant on $\big(\mathbb{P}^1, p_1,\dots , p_n\big)$ which are semistable with respect to $\mathcal{A}$. Under some conditions on the weights, we determine and give a modular interpretation for the automorphism group of the moduli space $\mathcal{M}_{\mathcal{A}}$. It is isomorphic to $\left(\frac{\mathbb{Z}}{2\mathbb{Z}}\right)^{k}$ for some $k\in \{0,\dots, n-1\}$, and is generated by admissible elementary transformations of parabolic vector bundles. The largest of these automorphism groups, with $k=n-1$, occurs for the central weight $\mathcal{A}_{F}= \left(\frac{1}{2},\dots,\frac{1}{2}\right)$. The corresponding moduli space ${\mathcal M}_{\mathcal{A}_F}$ is a Fano variety of dimension $n-3$, which is smooth if $n$ is odd, and has isolated singularities if $n$ is even.

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arXiv2013

On the order of the automorphism group of foliations

Let $\mathcal F$ be a holomorphic foliation with ample canonical bundle on a smooth projective surface $X$. We obtain an upper bound on the order of its automorphism group which depends only on $K_{\mathcal F}^2$ and $K_{\mathcal F}K_{X}$, provided this group is finite. Here, $K_{\mathcal F}$ and $K_{X}$ are the canonical bundles of $\mathcal F$ and $X$, respectively.

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arXiv2012

On the polar degree of projective hypersurfaces

Given a hypersurface in the complex projective $n$-space we prove several known formulas for the degree of its polar map by purely algebro-geometric methods. Furthermore, we give formulas for the degree of its polar map in terms of the degrees of the polar maps of its components. As an application, we classify the plane curves with polar map of low degree, including a very simple proof of I. Dolgachev's classification of homaloidal plane curves.

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arXiv2011

Characteristic Numbers and invariant subvarieties for Projective Webs

We define the characteristic numbers of a holomorphic k-distribution of any dimension on $mathbb P^n$ and obtain relations between these numbers and the characteristic numbers of an invariant subvariety. As an application we bound the degree of a smooth invariant hypersurface.

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arXiv2011

Logarithmic Hesse's Problem

We show that if a Laurent polynomial on the coordinate ring of the complex algebraic torus on n variables has vanishing logarithmic Hessian, then up to an automorphism of the torus, the Laurent polynomial depends on at most n-1 variables.

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arXiv2007

On the degree of Polar Transformations -- An approach through Logarithmic Foliations

We investigate the degree of the polar transformations associated to a certain class of multi-valued homogeneous functions. In particular we prove that the degree of the pre-image of generic linear spaces by a polar transformation associated to a homogeneous polynomial $F$ is determined by the zero locus of $F$. For zero dimensional-dimensional linear spaces this was conjecture by Dolgachev and proved by Dimca-Papadima using topological arguments. Our methods are algebro-geometric and rely on the study of the Gauss map of naturally associated logarithmic foliations.

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