Compactified Jacobians Research Articles

Stability conditions for line bundles on nodal curves

Abstract We introduce the abstract notion of a smoothable fine compactified Jacobian of a nodal curve, and of a family of nodal curves whose general element is smooth. Then we introduce the combinatorial notion of a stability assignment for line bundles and their degenerations. We prove that smoothable fine compactified Jacobians are in bijection with these stability assignments. We then turn our attention to 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 $\gcd (d+1-g, 2g-2)=1$ .

Compactified Jacobians as Mumford models

We show that relative compactified Jacobians of one-parameter smoothings of a nodal curve of genus g g are Mumford models of the generic fiber. Each such model is given by an admissible polytopal decomposition of the skeleton of the Jacobian. We describe the decompositions corresponding to compactified Jacobians explicitly in terms of the auxiliary stability data and find, in particular, that in degree g g there is a unique compactified Jacobian encoding slope stability, and it is induced by the tropical break divisor decomposition.

Geometry of Genus One Fine Compactified Universal Jacobians

Abstract 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 $\overline {{\mathcal {C}}}_{1,n} / \overline {{\mathcal {M}}}_{1,n}$ 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 $\overline {{\mathcal {J}}}^{d}_{1,n}$ and prove that their even cohomology is algebraic.

The direct image of generalized divisors and the Norm map between compactified Jacobians

The Norm map between Jacobians and the Prym scheme are widely used in the description of spectral data of G-Higgs pairs, for G varying among classical Lie group. The aim of this paper is to generalize the Norm map to compactified Jacobians and provide a compactification for the Prym scheme. Given a finite, flat morphism between curves (e.g. embeddable noetherian schemes of pure dimension 1), we first define the notion of direct and inverse image for generalized divisors and generalized line bundles. In the case when we deal with (possibly reducible, non-reduced) projective curves over a field and the codomain curve is smooth, we introduce the compactified Jacobians parametrizing torsion-free rank-1 sheaves and then we study the Norm and inverse image maps between compactified Jacobians. Finally, we introduce and study the Prym stack defined as the kernel of the Norm map.

A support theorem for Hilbert schemes of planar curves, II

We study the cohomology of Jacobians and Hilbert schemes of points on reduced and locally planar curves, which are however allowed to be singular and reducible. We show that the cohomologies of all Hilbert schemes of all subcurves are encoded in the cohomologies of the fine compactified Jacobians of connected subcurves, via the perverse Leray filtration. We also prove, along the way, a result of independent interest, giving sufficient conditions for smoothness of the total space of the relative compactified Jacobian of a family of locally planar curves.

Fourier–Mukai and autoduality for compactified Jacobians, II

To every reduced (projective) curve X with planar singularities one can associate many fine compactified Jacobians, depending on the choice of a polarization on X, which are birational (possibly non-isomorphic) Calabi-Yau projective varieties with locally complete intersection singularities. We define a Poincare' sheaf on the product of any two (possibly equal) fine compactified Jacobians of X and show that the integral transform with kernel the Poincare' sheaf is an equivalence of their derived categories. In particular, any two fine compactified Jacobians are derived equivalent. When applied to the same fine compactified Jacobian, one gets a Fourier-Mukai autoequivalence, which generalizes the classical result of Mukai for Jacobians of smooth curves (or more generally abelian varieties) and of Arinkin for compactified Jacobians of integral curves, thus providing further evidence for the classical limit of the geometric Langlands conjecture (as formulated by R. Donagi and T. Pantev). As a corollary of our main result, we prove an autoduality result for fine compactified Jacobians: there is a natural equivariant open embedding of the connected component of the scheme parametrizing rank-1 torsion-free sheaves on X into the connected component of the algebraic space parametrizing rank-1 torsion-free sheaves on a given fine compactified Jacobian of X.

Fine compactified Jacobians of reduced curves

To every singular reduced projective curve X X one can associate many fine compactified Jacobians, depending on the choice of a polarization on X X , each of which yields a modular compactification of a disjoint union of a finite number of copies of the generalized Jacobian of X X . We investigate the geometric properties of fine compactified Jacobians focusing on curves having locally planar singularities. We give examples of nodal curves admitting nonisomorphic (and even nonhomeomorphic over the field of complex numbers) fine compactified Jacobians. We study universal fine compactified Jacobians, which are relative fine compactified Jacobians over the semiuniversal deformation space of the curve X X . Finally, we investigate the existence of twisted Abel maps with values in suitable fine compactified Jacobians.

The local structure of compactified Jacobians

This paper studies the local geometry of compactified Jacobians constructed by Caporaso, Oda-Seshadri, Pandharipande, and Simpson. The main result is a presentation of the completed local ring of the compactified Jacobian of a nodal curve as an explicit ring of invariants described in terms of the dual graph of the curve. The authors have investigated the geometric and combinatorial properties of these rings in previous work, and consequences for compactified Jacobians are presented in this paper. Similar results are given for the local structure of the universal compactified Jacobian over the moduli space of stable curves.

A support theorem for Hilbert schemes of planar curves

Consider a family of integral complex locally planar curves whose relative Hilbert scheme of points is smooth. The decomposition theorem of Beilinson, Bernstein, and Deligne asserts that the pushforward of the constant sheaf on the relative Hilbert scheme splits as a direct sum of shifted semisimple perverse sheaves. We will show that no summand is supported in positive codimension. It follows that the perverse filtration on the cohomology of the compactified Jacobian of an integral plane curve encodes the cohomology of {\em all} Hilbert schemes of points on the curve. Globally, it follows that a family of such curves with smooth relative compactified Jacobian ("moduli space of D-branes'') in an irreducible curve class on a Calabi-Yau threefold will contribute equally to the BPS invariants in the formulation of Pandharipande and Thomas, and in the formulation of Hosono, Saito, and Takahashi.

Picard Bundles and Brill–Noether Loci in the Compactified Jacobian of a Nodal Curve

Picard Bundles and Brill–Noether Loci in the Compactified Jacobian of a Nodal Curve

Fine compactified Jacobians

AbstractWe study Esteves's fine compactified Jacobians for nodal curves. We give a proof of the fact that, for a one‐parameter regular local smoothing of a nodal curve X, the relative smooth locus of a relative fine compactified Jacobian is isomorphic to the Néron model of the Jacobian of the general fiber, and thus it provides a modular compactification of it. We show that each fine compactified Jacobian of X admits a stratification in terms of certain fine compactified Jacobians of partial normalizations of X and, moreover, that it can be realized as a quotient of the smooth locus of a suitable fine compactified Jacobian of the total blowup of X. Finally, we determine when a fine compactified Jacobian is isomorphic to the corresponding Oda‐Seshadri's coarse compactified Jacobian.

On Néron models of moduli spaces of theta characteristics

Let f : C → B be a smoothing of a stable curve C and S f ∗ be the moduli space of theta characteristics on the smooth fibers of f. We describe the Néron model N ( S f ∗ ) , in terms of combinatorial invariants of the dual graph of C. Furthermore, we provide a modular description of N ( S f ∗ ) and we construct an immersion ψ f : N ( S f ∗ ) ↪ J E σ , where J E σ is a suitable relative compactified Jacobian. We show that ψ f factors through the locus of J E σ parametrizing locally free rank-1 sheaves.

Compactified Jacobians and Torelli Map

We compare several constructions of compactified jacobians — using semistable sheaves, semistable projective curves, degenerations of abelian varieties, and combinatorics of cell decompositions — and show that they are equivalent. We give a detailed description of the “canonical compactified jacobian” in degree g − 1 . Finally, we explain how Kapranov’s compactification of configuration spaces can be understood as a toric analog of the extended Torelli map.

Global Matric Massey Products and the Compactified Jacobian of the E6-Singularity

In this paper we compute the compactified Jacobian of the singularity E6. By G. M. Greuel and H. Knörrer (1985, Math. Ann.270, 417–425) this singularity has only a finite number of isomorphism classes of rank 1 torsionfree modules. Using the theory of Matric Massey products, in an earlier work we computed the local formal moduli with its local versal family for each local module, and we studied the degeneracy of each local module. Here we prove a result showing how the local theory connects to the global theory; i.e., we prove that the morphism from the local formal moduli of a local module to the local ring at the point corresponding to the module on the compactified Jacobian is a smooth morphism. In the case where M=E6, i.e., the normalization, this morphism is an isomorphism. Thus the degeneracy (stratification) diagram for the compactified Jacobian can be found from the degeneracy of the normalization in the local case.