On Varieties of Lines on Linear Sections of Grassmannians

TL;DR

This paper studies the structure of lines on certain linear sections of Grassmannians, computes automorphism group orbits, and classifies their Fano properties, contributing to the understanding of high-index Fano manifolds.

Contribution

It computes automorphism group orbits on linear sections of Grassmannians and describes the variety of lines passing through fixed points, advancing classification of high-index Fano manifolds.

Findings

Automorphism groups have finitely many orbits.

Descriptions of line varieties through fixed points.

These Fano manifolds are not weakly 2-Fano.

Abstract

General linear sections of codimension 2 of the Grassmannians G(1,4) and G(1,5) appear in the classification of Fano manifolds of high index. Unlike Grassmannians, these manifolds are not homogeneous. Nevertheless, their automorphisms groups have finitely many orbits. In this work we first compute the orbits of these actions. Then we give a description of the variety of lines (under the Pl\"ucker embedding) passing through a fixed point in each orbit of the action. As an application we show that these Fano manifolds are not weakly 2-Fano, completing the classification of weakly 2-Fano manifolds of high index, initiated by Carolina Araujo and Ana-Maria Castravet.