Kernel Sheaf on Integral Nodal Curves

In this article we study the stability of Kernel sheaf obtained from a generating subspace of rank one torsion-free sheaf on an integral nodal curve.

💡 Research Summary

This paper, titled “Kernel Sheaf on Integral Nodal Curves,” investigates the stability of kernel sheaves constructed from generating subspaces of rank one torsion-free sheaves on singular algebraic curves with ordinary double points (nodes). The authors, Sura Tno Basu, Krishanu Dan, and Aanjaneya Rath, extend the well-studied theory of syzygy (or kernel) bundles from smooth projective varieties to the setting of integral nodal curves.

The central object of study is defined as follows: for an integral projective nodal curve X, a globally generated torsion-free sheaf E of rank one, and a generating subspace V ⊆ H⁰(X, E), one has a natural evaluation map V⊗O_X → E. Its kernel, denoted M*_V,E, is the “kernel sheaf.” The paper focuses on establishing conditions under which M*_V,E is a stable sheaf with respect to the natural polarization.

The paper’s methodology is an adaptation of techniques used by Mistretta for smooth curves. The core strategy involves constructing a parameter space D_b,s that encodes all potential data leading to a destabilizing subsheaf of M*_V,L (where L is a rank-one torsion-free sheaf). By meticulously computing the dimension of this parameter space and comparing it to the dimension of the Grassmannian Gr(c, H⁰(X, L))—which parameterizes the generating subspaces V of codimension c—the authors show that for a “generic” choice of V, no destabilizing subsheaf can exist, hence proving stability.

The main results are presented in two tiers, based on the number of nodes.

For curves with a single node (Section 3):

• Theorem 3.2: Let X be an integral projective curve with one ordinary double point of arithmetic genus g ≥ 2. Let L be a globally generated torsion-free sheaf of rank one and degree d > 2g + 2c, where 1 ≤ c ≤ g. Then, for a generic subspace V ⊆ H⁰(X, L) of codimension c that generates L, the kernel sheaf M*_V,L is stable.
• Theorem 3.3: In the borderline case d = 2g + 2c, M*_V,L is always semistable. Furthermore, if X is non-hyperelliptic, it is in fact stable.

The proofs for the single-node case are technically involved. A key step is relating a potential destabilizing subsheaf N of M*_V,L to another rank-one torsion-free sheaf F and a generating subspace W ⊆ H⁰(X, F) via a diagram chase (Lemma 2.2). The homomorphism sheaf E = Hom(F, L) plays a critical role. The analysis branches into cases depending on whether E is locally free (a line bundle) or not, which relates to the local types of F and L at the node. In each case, the authors use the normalization map π: Y → X (where Y is the smooth curve of genus g-1) to pull back a related sheaf to a line bundle on Y, apply Clifford’s theorem to bound h⁰(X, E), and then perform the dimension comparison for D_b,s.

For curves with multiple nodes (Section 4):

• Theorem 4.1: Let X be an integral projective curve of arithmetic genus g ≥ 3 with n nodes. Let L be a globally generated torsion-free sheaf of rank one and degree d > 10(g + c)/3, for 1 ≤ c ≤ g. Then, for a generic V ⊆ H⁰(X, L) of codimension c, M*_V,L is stable. Moreover, if the number of nodes n ≤ g - 2, then stability holds under the weaker condition d > 2g + 2c.

The multi-node analysis introduces additional complexity due to the “local type” of a torsion-free sheaf at each node (isomorphic to the local ring or its maximal ideal). This affects the degree of the homomorphism sheaf E, giving a range s ≤ deg E ≤ s + r. The authors construct a related sheaf F’ that is of local type one at all nodes, push it forward from the normalization Y, and apply Clifford’s theorem to the corresponding line bundle on Y. The stronger degree requirement d > 10(g+c)/3 arises from this more general bound. The improvement when n ≤ g-2 demonstrates how a smaller number of nodes relaxes the constraints from the Clifford bound.

Throughout the paper, the authors carefully handle the subtleties of working with torsion-free sheaves on singular curves, including duality properties (Lemma 2.1) and the relationship between local freeness and Hom sheaves (Lemma 3.1). The work provides a significant generalization of stability results for kernel bundles, moving from smooth curves to the rich and technically demanding world of nodal singularities, and establishes precise numerical thresholds for stability dependent on genus, degree, codimension, and the number of nodes.