Rationally Simply Connected Hypersurfaces in Orthogonal Grassmannians

TL;DR

This paper investigates the moduli space of rational curves in low degree hypersurfaces within orthogonal Grassmannians, establishing conditions under which these hypersurfaces are rationally simply connected.

Contribution

It proves rationally simply connectedness for general hypersurfaces in orthogonal Grassmannians under specific degree conditions, advancing understanding of their geometric properties.

Findings

Rationally simply connectedness holds under certain degree bounds.

Conditions relate hypersurface degree to Grassmannian parameters.

Results extend knowledge of rational curves in orthogonal Grassmannians.

Abstract

In this paper, we study the moduli space of rational curves in a general low degree hypersurface in the Orthogonal Grassmanian $OG(k,n+1)$ of $k$-dimensional isotropic subspaces of an $n+1$-dimensional vector space equipped with a symmetric, non-degenerate, bilinear form. We prove rationally simply connectedness for such a general hypersurface of degree $d$ where $d$ satisfies $n+1-8k-4\ge (3k-1)d^2-d$.