# Standard Embeddings of Smooth Schubert Varieties in Rational Homogeneous Manifolds of Picard Number 1

## TL;DR

This paper characterizes standard embeddings of smooth Schubert varieties within rational homogeneous manifolds of Picard number 1 using varieties of minimal rational tangents, focusing on specific nonhomogeneous cases.

## Contribution

It provides a new characterization of embeddings of smooth Schubert varieties in certain rational homogeneous manifolds via minimal rational tangents.

## Key findings

- Standard embeddings are characterized using minimal rational tangents.
- Focus on nonhomogeneous smooth Schubert varieties in symplectic Grassmannians.
- Analysis includes the 20-dimensional F4-homogeneous manifold.

## Abstract

Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomogeneous smooth Schubert varieties in symplectic Grassmannians and in the 20-dimensional $F_4$-homogeneous manifold associated to a short simple root.