Resonance of rank-two vector bundles over elliptic curves

Abstract: In this note, we study the resonance variety of rank-two vector bundles over an elliptic curve. Our approach is based on analyzing the flattening stratification of the resonance. We also investigate the linear section of the Grassmann variety $\operatorname{Gr}(2,n)$ from which the resonance is constructed through the lens of its corresponding flattening stratification.

• The paper introduces a formal characterization of resonance varieties for rank-two bundles over elliptic curves, distinguishing split and non-split cases using flattening stratification and Atiyah’s classification.
• It employs techniques from Brill-Noether theory and geometric embeddings to explicitly determine dimensions and irreducible components, including projective spaces and quadric surfaces.
• The findings shed light on vector bundle moduli spaces and suggest promising extensions to higher-rank bundles and higher-genus curves.

Resonance Varieties of Rank-Two Vector Bundles Over Elliptic Curves

Introduction and Context

This work provides a detailed geometric and algebraic study of the resonance varieties associated to rank-two vector bundles on elliptic curves. Resonance varieties, initially developed in the context of hyperplane arrangements, have become a critical tool for understanding the interplay between the geometry of vector bundles and cohomological support loci. The paper adopts a formalism due to Papadima and Suciu, where the resonance variety of a pair $(V, K)$, with $K \subseteq \bigwedge^2 V$, encapsulates cohomological jump loci and their supporting modules.

Building upon recent advances that relate resonance varieties to flattening stratifications and Brill-Noether theory, the author systematically analyzes the structure of resonance for both split and non-split rank-two bundles over genus-one curves, leveraging Atiyah's classification and a detailed understanding of saturated sub-line bundles and their embeddings.

Mathematical Structure of Resonance Varieties

The formal definition underpinning the analysis is the resonance variety

$$\mathcal{R}(V, K) := \{ [a] \in \mathbb{P} V^{\vee} : \exists b \in V^{\vee},\ 0 \neq a \wedge b \in K^{\perp} \}.$$

For a vector bundle $E$ over a curve $C$, setting $V = H^0(C, E)^{\vee}$ and $K^{\perp} = \ker d_2$ (where $d_2$ is the second determinant map) leads to the definition of the resonance variety of $E$ in the projectivization of global sections.

Crucial to the analysis is the flattening stratification $\mathcal{R}(E) = \coprod_d \mathcal{R}_d(E)$, where each stratum is associated to saturated subpencils (sub-line bundles with at least two global sections) of a given degree $K \subseteq \bigwedge^2 V$0. The stratification is closely related to Brill-Noether loci, allowing the fibers of the projection maps to be interpreted as images under section multiplication, giving rise to geometric structures such as Segre embeddings and Grassmannians.

Analysis for Rank-Two Bundles on Elliptic Curves

The paper separates the analysis into non-split and split rank-two bundles, reflecting the dichotomy from Atiyah's classification.

Non-Split Case

For non-split bundles, classified as either unique non-trivial extensions of line bundles of equal or differing degrees, the resonance variety is either empty (if $K \subseteq \bigwedge^2 V$1) or a projective space $K \subseteq \bigwedge^2 V$2, where $K \subseteq \bigwedge^2 V$3. The resonance is always linear and, for degree at least $K \subseteq \bigwedge^2 V$4, occupies a linear subspace of the ambient space of sections. There is precisely one saturated sub-line bundle, and hence the resonance variety consists solely of the projectivization of its sections.

Split Case

The split case, $K \subseteq \bigwedge^2 V$5, demonstrates far more intricate structure. The existence and enumeration of saturated sub-line bundles are governed by the specific arithmetic of $K \subseteq \bigwedge^2 V$6, $K \subseteq \bigwedge^2 V$7, possible isomorphism relations between $K \subseteq \bigwedge^2 V$8 and $K \subseteq \bigwedge^2 V$9, and the geometry of $$\mathcal{R}(V, K) := \{ [a] \in \mathbb{P} V^{\vee} : \exists b \in V^{\vee},\ 0 \neq a \wedge b \in K^{\perp} \}.$$0 as an elliptic curve.

• For $$\mathcal{R}(V, K) := \{ [a] \in \mathbb{P} V^{\vee} : \exists b \in V^{\vee},\ 0 \neq a \wedge b \in K^{\perp} \}.$$1, the resonance is either empty or a projective space of dimension $$\mathcal{R}(V, K) := \{ [a] \in \mathbb{P} V^{\vee} : \exists b \in V^{\vee},\ 0 \neq a \wedge b \in K^{\perp} \}.$$2 as only $$\mathcal{R}(V, K) := \{ [a] \in \mathbb{P} V^{\vee} : \exists b \in V^{\vee},\ 0 \neq a \wedge b \in K^{\perp} \}.$$3 may yield a subpencil.
• When $$\mathcal{R}(V, K) := \{ [a] \in \mathbb{P} V^{\vee} : \exists b \in V^{\vee},\ 0 \neq a \wedge b \in K^{\perp} \}.$$4, the resonance can be reducible and, in several cases (e.g., $$\mathcal{R}(V, K) := \{ [a] \in \mathbb{P} V^{\vee} : \exists b \in V^{\vee},\ 0 \neq a \wedge b \in K^{\perp} \}.$$5, $$\mathcal{R}(V, K) := \{ [a] \in \mathbb{P} V^{\vee} : \exists b \in V^{\vee},\ 0 \neq a \wedge b \in K^{\perp} \}.$$6), takes the form of the disjoint union of projective lines. For $$\mathcal{R}(V, K) := \{ [a] \in \mathbb{P} V^{\vee} : \exists b \in V^{\vee},\ 0 \neq a \wedge b \in K^{\perp} \}.$$7, the resonance becomes a smooth quadric surface, namely the image of the Segre embedding $$\mathcal{R}(V, K) := \{ [a] \in \mathbb{P} V^{\vee} : \exists b \in V^{\vee},\ 0 \neq a \wedge b \in K^{\perp} \}.$$8.
• In higher degree cases ($$\mathcal{R}(V, K) := \{ [a] \in \mathbb{P} V^{\vee} : \exists b \in V^{\vee},\ 0 \neq a \wedge b \in K^{\perp} \}.$$9), the resonance is dominated by the closure of the first non-empty stratum, typically $E$0. This closure becomes an irreducible projective variety of dimension $E$1 containing, in its closure, all higher strata and their images under the multiplication maps. The structure is shown to be connected and, aside from low-degree exceptions, irreducible.
• The description extends to the associated linear sections $E$2 of the Grassmannian $E$3, where the analysis of irreducible components and their dimensions closely parallels the resonance variety, confirming that in most split cases, reducibility is prevalent except for special symmetry (e.g., $E$4, $E$5).

Notable Explicit Results

• Non-split $E$6: $E$7 if $E$8; else $E$9 (where $C$0).
• Split $C$1 with $C$2: $C$3 is trivial or a projective space depending on $C$4.
• $C$5, $C$6: $C$7.
• $C$8: $C$9 is a smooth quadric $V = H^0(C, E)^{\vee}$0.
• General split $V = H^0(C, E)^{\vee}$1 with $V = H^0(C, E)^{\vee}$2: $V = H^0(C, E)^{\vee}$3 is a unique irreducible, connected component of maximal dimension $V = H^0(C, E)^{\vee}$4.
• Associated Grassmannians: The number of irreducible components of $V = H^0(C, E)^{\vee}$5 is generally at least the number of non-empty strata. $V = H^0(C, E)^{\vee}$6 is irreducible if and only if $V = H^0(C, E)^{\vee}$7, $V = H^0(C, E)^{\vee}$8, and $V = H^0(C, E)^{\vee}$9.

Implications and Future Directions

The theoretical implications of these results are multifold:

• The fine stratification and explicit characterization of resonance varieties for rank-two bundles on elliptic curves enhance the understanding of how subbundle geometry impacts support loci and, consequently, module-theoretic structures (such as Koszul modules) associated to the bundle.
• The reducibility and connectedness results answer subtle questions about how resonance varieties degenerate or become singular in the presence of symmetry or higher-dimensional cohomology.
• The techniques employed—including flattening stratification via Quot schemes, combinatorial isomorphisms related to divisor classes and Brill-Noether loci, and explicit computation of embeddings—provide a robust toolkit potentially extensible to higher genus or higher rank cases, even though the author notes that genus one is particularly tractable due to the richness of the Jacobian and explicit classification of bundles.

Practical implications, while more speculative, suggest that the structure of resonance varieties could inform the design and analysis of moduli spaces of vector bundles; for instance, in understanding where jump loci and degeneracy loci overlap or diverge. Connections to cohomological Hall algebras or to the study of syzygies (as in the Green's conjecture framework) are also suggested by the links to Koszul modules and support varieties.

For future research, two clear directions emerge:

• Extension of the flattening stratification analysis to bundles of higher rank or over base curves of higher genus, where the arithmetic of divisor classes and Brill-Noether loci becomes more complex.
• Detailed computation of irreducible components of resonance varieties and linear sections of Grassmannians for more general families of bundles—possibly including instability or further degeneracy—could yield further insights into global properties of moduli spaces.

Conclusion

The paper delivers a rigorous and explicit analysis of the resonance varieties of rank-two vector bundles over elliptic curves, elucidating the interplay between vector bundle geometry, cohomological jump loci, and algebraic support varieties. The results confirm that for elliptic curves, resonance varieties exhibit rich and tractable geometric structures, with precise criteria for irreducibility, connectivity, and dimension—distinguishing sharply between the split and non-split cases. The techniques and results pave the way for further exploration in more general settings, with potential implications for the global theory of vector bundles, moduli spaces, and related areas in algebraic geometry and representation theory.