[Paper Review] Degeneration of intermediate Jacobians and the Torelli theorem

Suratno Basu, Ananyo Dan+1 more|arXiv (Cornell University)|Sep 23, 2018

Algebraic Geometry and Number Theory34 references3 citations

TL;DR

This paper establishes a higher-rank Torelli theorem for irreducible nodal curves with one node, proving that such a curve is uniquely determined by the second intermediate Jacobian of its Gieseker moduli space of rank 2 semi-stable sheaves with fixed odd-degree determinant. Using degeneration of mixed Hodge structures and Néron models, the authors overcome the failure of pure Hodge theory in the singular case, extending Mumford–Newstead’s result to nodal curves under non-hyperelliptic conditions.

ABSTRACT

Mumford and Newstead generalized the classical Torelli theorem to higher rank i.e., a smooth, projective curve $X$ is uniquely determined by the second intermediate Jacobian of the moduli space of stable rank $2$ bundles on $X$, with fixed odd degree determinant. In this article we prove the analogous result in the case $X$ is an irreducible nodal curve with one node. As a byproduct, we obtain the degeneration of the second intermediate Jacobians and the associated Néron model of a family of such moduli spaces.

Motivation & Objective

• To extend the Mumford–Newstead higher-rank Torelli theorem to irreducible nodal curves with one node.
• To establish that the second intermediate Jacobian of the Gieseker moduli space determines the curve up to isomorphism in the nodal case.
• To analyze the degeneration of intermediate Jacobians and construct the associated Néron model for families of moduli spaces over nodal curves.
• To overcome the failure of pure Hodge structures in the singular setting by using limit mixed Hodge structures and non-trivial monodromy.

Proposed method

• Use of relative Gieseker moduli spaces over a family of curves degenerating to a nodal curve with one node.
• Construction of the second intermediate Jacobian of the central fiber moduli space using mixed Hodge structures on H^3.
• Application of Néron model theory to families of intermediate Jacobians to handle semi-abelian structures arising from singular fibers.
• Employment of limit mixed Hodge structures with non-trivial monodromy, distinguishing this case from previous pure Hodge structure approaches.
• Utilization of the universal family of curves over the moduli space and pullback of tautological bundles to analyze the geometry of the central fiber.
• Analysis of the central fiber as a reduced simple normal crossings divisor with two components intersecting in a P^1 × P^1-bundle over the moduli space of the normalization.

Experimental results

Research questions

• RQ1 Can the higher-rank Torelli theorem be extended to irreducible nodal curves with one node, analogous to the smooth case?
• RQ2 How do intermediate Jacobians degenerate in families of moduli spaces over nodal curves, and what structure do they acquire in the limit?
• RQ3 What role does non-trivial monodromy play in the limit mixed Hodge structure of the second intermediate Jacobian in the nodal case?
• RQ4 Can the Néron model of the intermediate Jacobian be constructed and used to recover the curve from its moduli space?
• RQ5 Why is the non-hyperelliptic condition necessary for the Torelli-type reconstruction in the nodal setting?

Key findings

• The second intermediate Jacobian of the Gieseker moduli space of rank 2 semi-stable sheaves with fixed odd-degree determinant is a semi-abelian variety in the nodal case.
• The central fiber of the relative Gieseker moduli space is a reduced simple normal crossings divisor with two irreducible components, one of which is isomorphic to a P^3-bundle over the moduli space of the normalization.
• The intersection of the two components is isomorphic to a P^1 × P^1-bundle over the moduli space of the normalization of the nodal curve.
• The limit mixed Hodge structure on H^3 of the central fiber is non-pure due to non-trivial monodromy, distinguishing this case from previous approaches with trivial monodromy.
• The Néron model of the intermediate Jacobian exists and captures the degeneration behavior of the family, enabling curve reconstruction.
• The curve is uniquely determined by its moduli space: if two such nodal curves have isomorphic moduli spaces, then the curves themselves are isomorphic, provided their normalizations are non-hyperelliptic.