# Projective aspects of the Geometry of Lagrangian Grassmannians and Spinor varieties

## TL;DR

This paper investigates the projective geometric properties of Lagrangian Grassmannians and Spinor varieties, focusing on osculating spaces and secant varieties, revealing smaller-than-expected osculating dimensions and conditions for secant non-defectivity.

## Contribution

It provides new insights into the osculating behavior and secant defectivity conditions of these varieties in their embeddings, advancing understanding of their projective geometry.

## Key findings

- Osculating dimension is smaller than expected for these varieties.
- Numerical conditions for secant non-defectivity are established.
- Results apply to both Plücker and Spinor embeddings.

## Abstract

We study the projective behavior, mainly with respect to osculating spaces and secant varieties, of Lagrangian Grassmannians and Spinor varieties. We prove that these varieties have osculating dimension smaller than expected. Furthermore, we give numerical conditions ensuring the non secant defectivity of Lagrangian Grassmannians in their Pl\"ucker embedding and of Spinor varieties in both their Pl\"ucker and Spinor embeddings.