arXiv subjects

Giulio Orecchia

Publications and source records attributed to Giulio Orecchia.

7 recordsLinked to original sources

arXiv2024

Logarithmic moduli of roots of line bundles on curves

We use the theory of logarithmic line bundles to construct compactifications of spaces of roots of a line bundle on a family of curves, generalising work of a number of authors. This runs via a study of the torsion in the tropical and logarithmic jacobians (recently constructed by Molcho and Wise). Our moduli space carries a `double ramification cycle' measuring the locus where the given root is isomorphic to the trivial bundle, and we give a tautological formula for this class in the language of piecewise polynomial functions (as recently developed by Molcho-Pandharipande-Schmitt and Holmes-Schwarz).

math.AG↗

arXiv2024

Models of Jacobians of curves

We show that the Jacobians of prestable curves over toroidal varieties always admit Néron models. These models are rarely quasi-compact or separated, but we also give a complete classification of quasi-compact separated group-models of such Jacobians. In particular we show the existence of a maximal quasi-compact separated group model, which we call the saturated model, which has the extension property for all torsion sections. The Néron model and the saturated model coincide over a Dedekind base, so the saturated model gives an alternative generalisation of the classical notion of Néron models to higher-dimensional bases; in the general case we give necessary and sufficient conditions for the Néron model and saturated model to coincide. The key result, from which most others descend, is that the logarithmic Jacobian of \cite{Molcho2018The-logarithmic} is a log Neron model of the Jacobian.

math.AG↗

arXiv2024

BPS invariants from $p$-adic integrals

We define $p$-adic BPS or $p$BPS-invariants for moduli spaces $M_{β,χ}$ of 1-dimensional sheaves on del Pezzo surfaces by means of integration over a non-archimedean local field $F$ . Our definition relies on a canonical measure $μ_{can}$ on the $F$-analytic manifold associated to $M_{β,χ}$ and the $p$BPS-invariants are integrals of natural $\mathbb{G}_m$-gerbes with respect to $μ_{can}$. A similar construction can be done for meromorphic Higgs bundles on a curve. Our main theorem is a $χ$-independence result for these $p$BPS-invariants. For 1-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement of $p$BPS with usual BPS-invariants trough a result of Maulik-Shen.

math.AG↗

arXiv2019

Unramified F-divided objects and the étale fundamental pro-groupoid in positive characteristic

Fix a scheme $S$ of characteristic $p$. Let $\mathscr{M}$ be an $S$-algebraic stack and let $\mbox{Fdiv}(\mathscr{M})$ be the stack of $\mbox{F}$-divided objects, that is sequences of objects $x_i\in\mathscr{M}$ with isomorphisms $σ_i:x_i\to \mbox{F}^*x_{i+1}$. Let $\mathscr{X}$ be a flat, finitely presented $S$-algebraic stack and $\mathscr{X}\to Π_1(\mathscr{X}/S)$ the étale fundamental pro-groupoid, constructed in the present text. We prove that if $\mathscr{M}$ is a quasi-separated Deligne-Mumford stack and $\mathscr{X}\to S$ has geometrically reduced fibres, there is a bifunctorial isomorphism of stacks \[\mathscr{H}\!om(Π_1(\mathscr{X}/S),\mathscr{M}) \simeq \mathscr{H}\!om(\mathscr{X},\mbox{Fdiv}(\mathscr{M})).\] In particular, the system of relative Frobenius morphisms $\mathscr{X}\to \mathscr{X}^{p/S}\to \mathscr{X}^{p^2/S}\to\dots$ allows to recover the space of connected components $π_0(\mathscr{X}/S)$ and the relative étale fundamental gerbe. In order to obtain these results, we study the existence and properties of relative perfection for algebras in characteristic $p$.

math.AG↗

arXiv2019

A monodromy criterion for existence of Neron models of abelian schemes in characteristic zero

We consider the problem of existence of Neron models for a family of abelian varieties over a base of dimension greater than 1. We show that when S is of equicharacteristic zero, the condition of toric additivity introduced in [Ore18] is sufficient for the existence of a Neron model, and also necessary under some extra assumptions. Furthermore, we give an equivalent formulation of toric additivity in terms of monodromy action on the l-adic Tate module.

math.AG↗

arXiv2018

A criterion for existence of Néron models of jacobians

Néron models of abelian varieties do not necessarily exist if the base $S$ has dimension higher than 1. We introduce a new condition, called toric additivity, on a family of smooth curves having nodal reduction over a normal crossing divisor $D\subset S$. The condition is necessary and sufficient for existence of a Néron model of the jacobian of the family; it depends only on the Betti numbers of the dual graphs of the fibres of the family, or on the toric ranks of the fibres of the jacobian.

math.AG↗

arXiv2016

Semi-factorial nodal curves and Néron models of jacobians

A family of curves over a discrete valuation ring is called semi-factorial if every line bundle on the generic fibre extends to a line bundle on the total space. In the nodal case, we give a sufficient and necessary condition for semi-factoriality, in terms of combinatorics of the dual graph of the special fibre. In particular, we show that performing one blow-up centered at the non-regular closed points yields a semi-factorial model of the generic fibre. As an application, we extend Raynaud's construction of the Néron (lft)-model of the jacobian of the generic fibre of a family of nodal curves to the case where the generic fibre is singular.

math.AG↗