arXiv subjects

Roberto Fringuelli

Publications and source records attributed to Roberto Fringuelli.

10 recordsLinked to original sources

arXiv2025

Hitchin fibrations are Ng\^{o} fibrations

We study the geometry of the Hitchin fibration for $\mathcal{L}$-valued $G$-Higgs bundles over a smooth projective curve of genus $g$, where $G$ is a reductive group and $\mathcal{L}$ is a suitably positive line bundle. We show that the Hitchin fibration admits the structure of a weak Abelian fibration. In the case when the line bundle $\mathcal{L}$ is a twist of the canonical bundle of the curve by a (possibly empty) reduced effective divisor, we prove a cohomological bound and $\delta$-regularity of the weak Abelian fibration.

math.AG↗

arXiv2024

The line bundles on the moduli stack of principal bundles on families of curves

Given a connected reductive algebraic group G, we investigate the Picard group of the moduli stack of principal G-bundles over an arbitrary family of smooth curves.

math.AG↗

arXiv2023

Integral Picard group of moduli of polarized K3 surfaces

We compute the integral Picard group of the moduli stack of polarized K3 surfaces of fixed degree whose singularities are at most rational double points. We also compute the integral Picard group of the stack of quasi-polarized K3 surfaces, and of the stacky period domain.

math.AG↗

arXiv2023

On the Picard group scheme of the moduli stack of stable pointed curves

The aim of the present paper is to study the (abstract) Picard group and the Picard group scheme of the moduli stack of stable pointed curves over an arbitrary scheme. As a byproduct, we compute the Picard groups of the moduli stack of stable or smooth pointed curves over a field of characteristic different from two.

math.AG↗

arXiv2022

The Picard group of the universal moduli stack of principal bundles on pointed smooth curves

For any smooth connected linear algebraic group G over an algebraically closed field k, we describe the Picard group of the universal moduli stack of principal G-bundles over pointed smooth k-projective curves.

math.AG↗

arXiv2021

Automorphisms of moduli spaces of principal bundles over a smooth curve

For any almost-simple group $G$ over an algebraically closed field $k$ of characteristic zero, we describe the automorphism group of the moduli space of semistable $G$-bundles over a connected smooth projective curve $C$ of genus at least $4$. The result is achieved by studying the singular fibers of the Hitchin fibration. As a byproduct, we provide a description of the irreducible components of two natural closed subsets in the Hitchin basis: the divisor of singular cameral curves and the divisor of singular Hitchin fibers.

math.AG↗

arXiv2020

The Picard group of the universal moduli stack of principal bundles on pointed smooth curves II

In this paper, which is a sequel of arXiv:2002.07494, we investigate, for any reductive group $G$ over an algebraically closed field $k$, the Picard group of the universal moduli stack $\mathrm{Bun}_{G,g,n}$ of $G$-bundles over $n$-pointed smooth projective curves of genus $g$. In particular: we give new functorial presentations of the Picard group of $\mathrm{Bun}_{G,g,n}$; we study the restriction homomorphism onto the Picard group of the moduli stack of principal $G$-bundles over a fixed smooth curve; we determine the Picard group of the rigidification of $\mathrm{Bun}_{G,g,n}$ by the center of $G$ as well as the image of the obstruction homomorphism of the associated gerbe. As a consequence, we compute the divisor class group of the moduli space of semistable $G$-bundles over $n$-pointed smooth projective curves of genus $g$.

math.AG↗

arXiv2018

The Brauer group of the universal moduli space of vector bundles over smooth curves

We compute the Brauer group of the universal moduli stack of vector bundles on (possibly marked) smooth curves of genus at least three over the complex numbers. As consequence, we obtain an explicit description of the Brauer group of the smooth locus of the associated moduli space of semistable vector bundles, when the genus is at least four.

math.AG↗

arXiv2018

The Picard group of the universal moduli space of vector bundles on stable curves

We construct the moduli stack of properly balanced vector bundles on semistable curves and we determine explicitly its Picard group. As a consequence, we obtain an explicit description of the Picard groups of the universal moduli stack of vector bundles on smooth curves and of the Schmitt's compactification over the stack of stable curves. We prove some results about the gerbe structure of the universal moduli stack over its rigidification by the natural action of the multiplicative group. In particular, we give necessary and sufficient conditions for the existence of Poincaré bundles over the universal curve of an open substack of the rigidification, generalizing a result of Mestrano-Ramanan.

math.AG↗

arXiv2016

The Picard group of the universal abelian variety and the Franchetta conjecture for abelian varieties

We compute the Picard group of the universal abelian variety over the moduli stack $\mathscr A_{g,n}$ of principally polarized abelian varieties over $\mathbb{C}$ with a symplectic principal level $n$-structure. We then prove that over $\mathbb{C}$ the statement of the Franchetta conjecture holds in a suitable form for $\mathscr A_{g,n}$.

math.AG↗