Subvarieties of hypersurface sections of generalized Grassmannians

Abstract

Consider a generalized Grassmannian $G/P\subset \mathbb{P}$ embedded in a projective space by a complete linear system of a positive generator of the Picard group. For a very general hypersurface $\mathcal{H}\subset \mathbb{P}$, we study subvarieties of $G/P\cap \mathcal{H}$ that are not of general type (or not of positive geometric genus). When the degree of the hypersurface is not small, we show that, under a certain condition on the parabolic subgroup P, such subvarieties are union of lines. Our result is a generalization of Clemens-Ran’s result concerning $G/P={\mathbb{P}}^n$.

KEYWORDS:

• General points
• Grassmannians
• isomorphisms

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 14J70
• 14C99
• 14M17
• 32Q45

Additional information

Funding

This work was supported by JSPS KAKENHI Grant Number 18K03246 and 22K03232.