Isotropic Quot schemes of orthogonal bundles over a curve

TL;DR

This paper investigates the structure and properties of isotropic Quot schemes of orthogonal bundles over a curve, revealing their irreducibility, emptiness conditions, and the nature of their saturated and nonsaturated parts.

Contribution

It provides a detailed analysis of the irreducible components and saturation properties of isotropic Quot schemes for orthogonal bundles, including new characterizations based on topological types.

Findings

For certain topological types, $IQ_e(V)$ is empty for all $e$.

In some cases, $IQ_e(V)$ contains irreducible components of nonsaturated subsheaves.

The closure of $IQ^o_e(V)$ has at most two irreducible components for $e o - abla$.

Abstract

We study the isotropic Quot schemes $IQ_e (V)$ parameterizing degree $e$ isotropic subsheaves of maximal rank of an orthogonal bundle $V$ over a curve. The scheme $IQ_e (V)$ contains a compactification of the space $IQ^o_e (V)$ of degree $e$ maximal isotropic subbundles, but behaves quite differently from the classical Quot scheme, and the Lagrangian Quot scheme in [6]. We observe that for certain topological types of $V$, the scheme $IQ_e (V)$ is empty for all $e$. In the remaining cases, for infinitely many $e$ there are irreducible components of $IQ_e (V)$ consisting entirely of nonsaturated subsheaves, and so $IQ_e (V)$ is strictly larger than the closure of $IQ^o_e (V)$. As our main result, we prove that for any orthogonal bundle $V$ and for $e \ll 0$, the closure $\overline{IQ^o_e (V)}$ of $IQ^o_e (V)$ 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 $\overline{IQ^o_e (V)}$ when $V$ has even rank.