Compactified symplectic leaves in bundle moduli spaces

Let ℰ be a rank-2 vector bundle over an elliptic curve E, decomposable as a sum of line bundles of degrees d' > d ≥ 2, and L the determinant of ℰ. The subspace L(ℰ) ⊂ ℙⁿ⁻¹ ≅ ℙExt¹(L, O_E) consisting of classes of extensions with middle term isomorphic to ℰ is one of the symplectic leaves of a remarkable Poisson structure on ℙⁿ⁻¹ defined by Feigin–Odesskii and Polishchuk, and all symplectic leaves arise in this manner, as shown in earlier work that realizes L(ℰ) as the base space of a principal Aut(ℰ)-fibration. Here, we embed L(ℰ) into a larger, projective base space L(E) of a principal Aut(ℰ)-fibration whose total space parametrizes sections of ℰ. The embedding realizes L(E)⊂L(E) as a complement of an anticanonical divisor Y (one of the main results), and we give an explicit description of the normalization of Y as a projective-space bundle over a projective space. For d=2,L(E) is one of the three Hirzebruch surfaces Σ_i, i = 0, 1, 2; we determine which occurs when and hence also the cases when L(ℰ) is affine. Separately, we prove that for d < n/2 the singular locus of the secant slice Sec_d,z (E) ⊂ ℙⁿ⁻¹, the portion of the d^th secant variety of E consisting of points lying on spans of d-tuples with sum z ∈ E, is precisely Sec_d−2. This strengthens the result that L(ℰ) is smooth, appearing in prior joint work with R. Kanda and S. P. Smith.