Finite incidence models for elliptic resonance varieties

Let E = A ⊕ B be a sum of line bundles of degrees 2 ≤ a ≤ b on a smooth complex elliptic curve. We determine all irreducible components of the locus of two-dimensional spaces of sections that generate a sheaf of rank one. Their number is a if A ≇ B, and max{1, a−2} if A ≅ B. Relative Grassmannians and projective bundles give smooth finite normalization models for these components. In saturation degree two, the normalization maps are closed embeddings.

For sections, elliptic resonance is the classical locus of sections with at least two zeros. Its standard incidence over the symmetric square therefore gives irreducibility whenever the minimal Harder–Narasimhan slope μ_min(E) exceeds two, in any rank at least two. In rank two, this follows from Bielawski’s two-zero incidence and answers Spiridon’s component question. For split bundles with 3 ≤ a ≤ b, we show that the incidence is the normalization of the section resonance, except for the diagonal degree-three case, where it is a cover of degree three. For a nonsplit rank-two bundle of degree e, section resonance is empty if e < 4, a projective line if e = 4, and irreducible of dimension e−3 if e ≥ 5. This corrects the proposed projective-space description in higher degree.

GitHub repository: https://github.com/eulogica/elliptic-resonance-revision