arXiv subjects

Nicola Pagani

Publications and source records attributed to Nicola Pagani.

At least 19 recordsLinked to original sources

arXiv2026

Extending Andreotti's proof of Torelli's theorem to nodal curves

Let $C$ be a connected nodal curve of genus $g$ over an algebraically closed field $k$ of characteristic zero. Let \[ A(C)=\Big(J(C), {\overline{P^{g-1}_C}}, Θ(C) \Big) \] be the stable semi-abelic pair (in the sense of Alexeev) associated with its canonical compactified Jacobian $\overline{P^{g-1}_C}$ in degree $g-1$. The canonical triple of $C$ consists of the closure of its canonical image together with two sets of distinguished points encoding the information of the images of the nonseparating nodes and of the branch data of the canonical map. We prove a functorial reconstruction: given connected nodal curves $C$ and $C'$ over $k$, every isomorphism $A(C)\cong A(C')$ of stable semi-abelic pairs canonically induces a projectivity between the corresponding canonical triples of $C$ and $C'$. Passing to isomorphism classes, this recovers the reconstruction statement in the compactified Torelli theorem of Caporaso-Viviani. Our proof extends Andreotti's Gauss-map strategy. The branch locus of the projection from the normalization of the Gauss graph recovers the nonlinear canonical components and branch data by projective biduality; boundary strata of the compactified Jacobian recover the node-images; and incidence arguments recover the linear components. Along the way, we repair technical gaps in the classical treatments of Andreotti's proof of Torelli's theorem for smooth curves.

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arXiv2026

A complete classification of modular compactifications of the universal Jacobian

This is the third paper in a series, following [FPVa] and [FPVb]. We classify all modular compactifications of the universal Jacobian over $\overline{\mathcal{M}}_{g,n}$, both as stacks and as their relative good moduli spaces. Our main result gives a combinatorial parametrization of compactified universal Jacobian stacks by $V$-functions on a stability domain $\mathbb{D}_{g,n}$ of half-vine types (two-components topological types with a chosen side); under this correspondence, fine compactifications are exactly the general $V$-functions. We single out the classical compactified universal Jacobians, namely those induced by numerical polarizations (relative $\mathbb{R}$-line bundles on the universal curve $\overline{\mathcal{C}}_{g,n}/\overline{\mathcal{M}}_{g,n}$), recovering the constructions of Kass-Pagani and Melo in the fine case, and we prove that their good moduli spaces are locally projective over $\overline{\mathcal{M}}_{g,n}$. We determine when two compactified universal Jacobians are isomorphic over $\overline{\mathcal{M}}_{g,n}$ and describe a resolution of the universal family via a compactified Jacobian over $\overline{\mathcal{M}}_{g,n+1}$. Finally, we analyse the poset $Σ_{g,n}$ of compactified universal Jacobians, an extension of the poset of regions of the hyperplane arrangement of classical stability conditions $\mathcal{A}_{g,n}$ studied in Kass-Pagani. We prove that for $n=0$ all compactified universal Jacobians are those constructed by Caporaso. We then give an explicit description of the submaximal elements of $Σ_{g,n}$ for all $n$, generalizing the stability walls in the classical stability space $\mathcal{A}_{g,n}$ from Kass-Pagani's work.

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arXiv2025

Wall-crossing of universal Brill-Noether classes

We give an explicit graph formula, in terms of decorated boundary strata classes, for the wall-crossing of universal Brill-Noether classes. More precisely, fix n>0 and d<g , and two stability conditions ϕ^-, ϕ^+ for degree d compactified universal (over the moduli space of stable n-pointed curves of genus g) Jacobians that lie on opposite sides of a stability hyperplane. Our main result is a formula for the difference between the Brill-Noether classes, compared via the pullback along the (rational) identity map. The calculation involves constructing a resolution of the identity map by means of subsequent blow-ups.

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arXiv2025

A complete theory of smoothable compactified Jacobians of nodal curves

We introduce and study a new class of compactified Jacobians for nodal curves, that we call compactified Jacobians of vine type, or simply V-compactified Jacobians. This class is strictly larger than the class of classical compactified Jacobians, as constructed by Oda-Seshadri, Simpson, Caporaso and Esteves. We characterize V-compactified Jacobians as the compactified Jacobians that can arise as limits of Jacobians of smooth curves under a one-parameter smoothing of the nodal curve, extending previous works on fine compactified Jacobians by Pagani-Tommasi and Viviani to the case of all compactified Jacobians.

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arXiv2025

A new class of compactified Jacobians for families of reduced curves

This is the first paper of a series of three. Here we give an abstract definition of the relative compactified Jacobian of a family of reduced curves. We prove that, under some mild assumptions on the family of curves, the fibres of the relative Jacobian are schemes (and not just algebraic spaces). We define V-stability conditions, and use them to construct relative compactified Jacobians. This extends the classical methods to produce modular compactifications of the Jacobian. To conclude, we show that, in the case when the curves have at worst planar singularities, the compactified Jacobians constructed from V-stability conditions have the same good properties of the classical ones.

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arXiv2024

Stability conditions for line bundles on nodal curves

We introduce the abstract notion of a \emph{smoothable fine compactified Jacobian} of a nodal curve, and of a family of nodal curves whose general element is smooth. Then we introduce the notion of a combinatorial stability condition for line bundles and their degenerations. We prove that smoothable fine compactified Jacobians are in bijection with these stability conditions. We then turn our attention to \emph{fine compactified universal Jacobians}, that is, fine compactified Jacobians for the moduli space $\overline{\mathcal{M}}_g$ of stable curves (without marked points). We prove that every fine compactified universal Jacobian is isomorphic to the one first constructed by Caporaso, Pandharipande and Simpson in the nineties. In particular, without marked points, there exists no fine compactified universal Jacobian unless $\gcd(d+1-g, 2g-2)=1$.

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arXiv2022

Geometry of genus one fine compactified universal Jacobians

We introduce a general abstract notion of fine compactified Jacobian for nodal curves of arbitrary genus. We focus on genus 1 and prove combinatorial classification results for fine compactified Jacobians in the case of a single nodal curve and in the case of the universal family over the moduli space of stable pointed curves. We show that if the fine compactified Jacobian of a nodal curve of genus 1 can be extended to a smoothing of the curve, then it can be described as the moduli space of stable sheaves with respect to some polarisation. In the universal case we construct new examples of genus 1 fine compactified universal Jacobians. Then we give a formula for the Hodge and Betti numbers of each genus 1 fine compactified universal Jacobian and prove that their even cohomology is algebraic.

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arXiv2021

The stability space of compactified universal Jacobians

In this paper we describe compactified universal Jacobians, i.e. compactifications of the moduli space of line bundles on smooth curves obtained as moduli spaces of rank 1 torsion-free sheaves on stable curves, using an approach due to Oda-Seshadri. We focus on the combinatorics of the stability conditions used to define compactified universal Jacobians. We explicitly describe an affine space, the stability space, with a decomposition into polytopes such that each polytope corresponds to a proper Deligne-Mumford stack that compactifies the moduli space of line bundles. We apply this description to describe the set of isomorphism classes of compactified universal Jacobians (answering a question of Melo), and to resolve the indeterminacy of the Abel-Jacobi sections (addressing a problem raised by Grushevsky-Zakharov).

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arXiv2019

Extending the Double Ramification Cycle using Jacobians

We prove that the extension of the double ramification cycle defined by the first-named author (using modifications of the stack of stable curves) coincides with that defined by the last-two named authors (using an extended Brill-Noether locus on suitable compactified universal Jacobians). In particular, in the untwisted case we deduce that both of these extensions coincide with that constructed by Li and Graber-Vakil using a virtual fundamental class on a space of rubber maps.

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arXiv2019

Pullbacks of universal Brill-Noether classes via Abel-Jacobi morphisms

Following Mumford and Chiodo, we compute the Chern character of the derived pushforward $\textrm{ch} (R^\bulletπ_\ast\mathscr{O}(\mathsf{D}))$, for $\mathsf D$ an arbitrary element of the Picard group of the universal curve over the moduli stack of stable marked curves. This allows us to express the pullback of universal Brill-Noether classes via Abel-Jacobi sections to the compactified universal Jacobians, for all compactifications such that the section is a well-defined morphism.

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arXiv2017

Extensions of the universal theta divisor

The Jacobian varieties of smooth curves fit together to form a family, the universal Jacobian, over the moduli space of smooth marked curves, and the theta divisors of these curves form a divisor in the universal Jacobian. In this paper we describe how to extend these families over the moduli space of stable marked curves (or rather an open subset thereof) using a stability parameter. We then prove a wall-crossing formula describing how the theta divisor varies with the stability parameter. We use that result to analyze a divisor on the moduli space of smooth marked curves that has recently been studied by Grushevsky-Zakharov, Hain and Müller. In particular, we compute the pullback of the theta divisor studied in Alexeev's work on stable abelic varieties and in Caporaso's work on theta divisors of compactified Jacobians.

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arXiv2015

Moduli of abelian covers of elliptic curves

For any finite abelian group G, we study the moduli space of abelian $G$-covers of elliptic curves, in particular identifying the irreducible components of the moduli space. We prove that, in the totally ramified case, the moduli space has trivial rational Picard group, and it is birational to the moduli space M_{1,n}, where n is the number of branch points. In the particular case of moduli of bielliptic curves, we also prove that the boundary divisors are a basis of the rational Picard group of the admissible covers compactification of the moduli space. Our methods are entirely algebro-geometric.

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arXiv2014

Harer stability and orbifold cohomology

In this paper we review the combinatorics of the twisted sectors of $\mathcal{M}_{g,n}$, and we exhibit a formula for the age of each of them in terms of the combinatorial data. Then we show that orbifold cohomology of $\mathcal{M}_{g,n}$ when $g \to \infty$ reduces to its ordinary cohomology. We do this by showing that the twisted sector with minimal age is always the hyperelliptic twisted sector with all markings in the Weierstrass points; the age of the latter moduli space is just half its codimension in $\mathcal{M}_{g,n}$.

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arXiv2014

The class of the bielliptic locus in genus 3

Let the bielliptic locus be the closure in the moduli space of stable curves of the locus of smooth curves that are double covers of genus 1 curves. In this paper we compute the class of the bielliptic locus in \bar{M}_3 in terms of a standard basis of the rational Chow group of codimension-2 classes in the moduli space. Our method is to test the class on the hyperelliptic locus: this gives the desired result up to two free parameters, which are then determined by intersecting the locus with two surfaces in \bar{M}_3.

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arXiv2013

The orbifold cohomology of moduli of hyperelliptic curves

We study the inertia stack of [M_{0,n}/S_n], the quotient stack of the moduli space of smooth genus 0 curves with n marked points via the action of the symmetric group S_n. Then we see how from this analysis we can obtain a description of the inertia stack of H_g, the moduli stack of hyperelliptic curves of genus g. From this, we can compute additively the Chen-Ruan (or orbifold) cohomology of H_g.

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arXiv2013

The orbifold cohomology of moduli of genus 3 curves

In this work we study the additive orbifold cohomology of the moduli stack of smooth genus g curves. We show that this problem reduces to investigating the rational cohomology of moduli spaces of cyclic covers of curves where the genus of the covering curve is g. Then we work out the case of genus g=3. Furthermore, we determine the part of the orbifold cohomology of the Deligne-Mumford compactification of the moduli space of genus 3 curves that comes from the Zariski closure of the inertia stack of M_3.

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arXiv2012

Chen-Ruan cohomology of M_{1,n} and \bar{M}_{1,n}

In this work we compute the Chen--Ruan cohomology and the stringy Chow ring of the moduli spaces of smooth and stable $n$-pointed curves of genus 1. We suggest a definition for an Orbifold Tautological Ring in genus 1, which is both a subring of the Chen--Ruan cohomology and of the stringy Chow ring.

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arXiv2012

The Chen-Ruan cohomology of moduli of curves of genus 2 with marked points

In this work we describe the Chen-Ruan cohomology of the moduli stacks of smooth and stable genus 2 pointed curves, and its algebraic counterpart: the stringy Chow ring. In the first half of the paper we compute the additive structure of the Chen-Ruan cohomology ring for the moduli stack of stable pointed genus 2 curves, describing it as a rationally graded vector space. In the second part we give generators for the even Chen--Ruan cohomology ring as an algebra on the ordinary cohomology.

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