Compactified symplectic leaves in bundle moduli spaces

Abstract

Let $\mathcal{E}$ be a rank-2 vector bundle over an elliptic curve $E$, decomposable as a sum of line bundles of degrees $d^{\prime }>d\ge 2$, and $\mathcal{L}$ the determinant of $\mathcal{E}$. The subspace $L(\mathcal{E})\subset {\mathbb{P}}^{n-1}\cong \mathbb{P}{\mathrm{Ext}}^1(\mathcal{L},{\mathcal{O}}_E)$ consisting of classes of extensions with middle term isomorphic to $\mathcal{E}$ is one of the symplectic leaves of a remarkable Poisson structure on ${\mathbb{P}}^{n-1}$ defined by Feigin-Odesskii/Polishchuk, and all symplectic leaves arise in this manner, as shown in earlier work that realizes $L(\mathcal{E})$ as the base space of a principal $\mathrm{Aut}(\mathcal{E})$-fibration. Here, we embed $L(\mathcal{E})$ into a larger, projective base space $\tilde{L}(\mathcal{E})$ of a principal $\mathrm{Aut}(\mathcal{E})$-fibration whose total space consists of sections of $\mathcal{E}$. The embedding realizes $L(\mathcal{E})\subset \tilde{L}(\mathcal{E})$ 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 $d=2$ $\tilde{L}(\mathcal{E})$ is one of the three Hirzebruch surfaces ${\Sigma }_i$, $i=0,1,2$; we determine which occurs when and hence also the cases when $L(\mathcal{E})$ is affine. Separately, we prove that for $d<\frac{n}{2}$ the singular locus of the secant slice ${\mathrm{Sec}}_{d,z}(E)\subset {\mathbb{P}}^{n-1}$, the portion of the $d^{th}$ secant variety of $E$ consisting of points lying on spans of $d$-tuples with sum $z\in E$, is precisely ${\mathrm{Sec}}_{d-2}$. This strengthens result that $L(\mathcal{E})$ is smooth, appearing in prior joint work with R. Kanda and S.P. Smith.

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