Compactified symplectic leaves in bundle moduli spaces

Abstract Let ℰ be a rank-2 vector bundle over an elliptic curve E , decomposable as a sum of line bundles of degrees d ' &gt; d ≥ 2, and ℒ the determinant of ℰ. The subspace L (ℰ) ⊂ ℙ n −1 ≅ ℙExt 1 (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 ℙ n −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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover> <m:mi>L</m:mi> <m:mo>˜</m:mo> </m:mover> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">E</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:math> $\widetilde{L}(\mathcal{E})$ of a principal Aut(ℰ)-fibration whose total space parametrizes sections of ℰ. The embedding realizes <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>L</m:mi> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">E</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> <m:mo>⊂</m:mo> <m:mover> <m:mi>L</m:mi> <m:mo>˜</m:mo> </m:mover> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">E</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:math> $L(\mathcal{E})\subset \widetilde{L}(\mathcal{E})$ 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>d</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mspace width="thinmathspace"/> <m:mover> <m:mi>L</m:mi> <m:mo>˜</m:mo> </m:mover> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">E</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:math> $d=2,\, \widetilde{L}(\mathcal{E})$ 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 &lt; n /2 the singular locus of the secant slice Sec d , z ( E ) ⊂ ℙ n −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.