We study the isotropic Quot schemes IQe(V ) parametrizing degree e isotropic subsheaves of maximal rank of an orthogonal bundle V over a curve. The scheme IQe(V ) contains a compactification of the space IQe∘(V ) of degree e maximal isotropic subbundles, but behaves quite differently from the classical Quot scheme, and the Lagrangian Quot scheme in [D. Cheong, I. Choe and G. H. Hitching, Irreducibility of Lagrangian Quot schemes over an algebraic curve, preprint (2019), arXiv:1804.00052, v2]. We observe that for certain topological types of V, the scheme IQe(V ) is empty for all e. In the remaining cases, for infinitely many e there are irreducible components of IQe(V ) consisting entirely of nonsaturated subsheaves, and so IQe(V ) is strictly larger than the closure of IQe∘(V ). As our main result, we prove that for any orthogonal bundle V and for e ≪ 0, the closure IQe∘(V )¯ of IQe∘(V ) is either empty or consists of one or two irreducible connected components, depending on deg(V ) and e. In so doing, we also characterize the nonsaturated part of IQe∘(V )¯ when V has even rank.

中文翻译：

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曲线上正交束的各向同性报价方案

我们研究各向同性报价方案智商e(五 )参数化度e正交丛最大秩的各向同性子滑轮五在一条曲线上。方案智商e(五 )包含空间的紧凑化智商e∘(五 )学位e最大各向同性子束，但其行为与经典 Quot 方案和 [D. Cheong, I. Choe 和 GH Hitching，代数曲线上拉格朗日报价方案的不可约性，预印本 (2019)，arXiv:1804.00052, v2]。我们观察到对于某些拓扑类型五， 方案智商e(五 )对所有人都是空的e. 在其余情况下，对于无限多e有不可约的成分智商e(五 )完全由非饱和子滑轮组成，因此智商e(五 )严格大于闭包智商e∘(五 ). 作为我们的主要结果，我们证明对于任何正交丛五并且对于e ≪ 0, 闭包智商e∘(五 )¯的智商e∘(五 )要么是空的，要么由一个或两个不可约的连通分量组成，具体取决于度(五 )和e. 在这样做的过程中，我们还表征了非饱和部分智商e∘(五 )¯什么时候五甚至有排名。