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 $Σ_i$, $i=0,1,2$; 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.
Automorphisms and symplectic leaves of Calogero-Moser spaces
研究了由复反射群$W$的正规化子的有穷级元诱导的Calogero-Moser空间的自同构的不动点的子簇的辛叶.我们给出了它的辛叶的参数化{ \it à la Harish-Chandra}（推广了Bellamy和Losev的早期工作）。这个结果的灵感来自Calogero-Moser空间的几何和有限约化群的幂幺表示之间的神秘关系，这将是即将发表的论文的主题。
对于特征为零的代数闭域上的每个Poisson代数簇X，我们正则地在X上附加一个右D-模M（X）。如果X是仿射的，则M（X）在X上的代数分布空间中的解是X上的泊松迹，即，Hamiltonian流下不变的分布。当X有n个辛叶时，我们证明了M（X）是完整的。因此，当X是仿射的，并且有2/4个辛叶时，X上的泊松迹空间是有限维的。作为应用，我们证明了非交换的滤过代数代数，如果其关联的分次代数代数是Poisson簇的坐标环，且该坐标环具有2/4个辛叶，则该代数具有2/4个不可约的有限维表示. Ivan Losev的附录加强了这一点，表明在这样的代数中，有许多素理想，并且它们都是本原的。更一般地，对于任意态射φ：X -> Y和X上Poisson模N的拟相干层，我们在X上附加了一个右D-模M_φ（X，N），并证明了它是完整的，如果X有n个辛叶，φ是有限的，N是相干的.作为一个应用，非交换滤过代数的不可约表示的有限性结果推广到了相关的分次代数不一定是交换的，而是作为其中心上的模生成的情形，该中心是一个Poisson簇的坐标环，该Poisson簇具有100个辛叶.我们还明确地描述（在仿射簇和紧凑的光滑流形的设置）的空间的泊松迹在X=V/G，其中V是辛和G是一个有限群忠实地作用于V。特别是，我们表明，这个空间是有限维的。
