Compactified symplectic leaves in bundle moduli spaces

TL;DR

This paper studies symplectic leaves in bundle moduli spaces over elliptic curves, embedding them into larger projective spaces, describing their geometry, and analyzing secant varieties' singularities.

Contribution

It introduces an embedding of symplectic leaves into larger projective spaces, describes the normalization of divisors, and characterizes singularities of secant slices.

Findings

Embedded symplectic leaves as complements of anticanonical divisors.

Explicit normalization of the divisors as projective-space bundles.

Identified the singular locus of secant slices as lower secant varieties.

Abstract

Let $\mathcal{E}$ be a rank-2 vector bundle over an elliptic curve $E$, decomposable as a sum of line bundles of degrees $d'>d\ge 2$, 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 $\widetilde{L}(\mathcal{E})$ of a principal $\mathrm{Aut}(\mathcal{E})$-fibration whose total space consists of sections of $\mathcal{E}$. The embedding realizes $L(\mathcal{E})\subset \widetilde{L}(\mathcal{E})$ 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 $d=2$ $\widetilde{L}(\mathcal{E})$ 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 $d<\frac n2$ 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.