Spherical Geometry of Hilbert Schemes of Conics in Adjoint Varieties

TL;DR

This paper investigates the geometric structure of the Hilbert scheme of smooth conics in certain adjoint varieties, revealing their spherical nature and describing their conjugacy classes using contact geometry.

Contribution

It establishes that the normalization of the Hilbert scheme component is a spherical variety and computes its colored fan, providing new insights into the geometry of conics in adjoint varieties.

Findings

Normalization of the Hilbert scheme component is spherical.

Computed the colored fan of the normalization.

Described conjugacy classes of conics in adjoint varieties.

Abstract

For each adjoint variety not of type $A$ or $C$, we study the irreducible component of the Hilbert scheme which parametrizes all smooth conics. We prove that its normalization is a spherical variety by using contact geometry, and then compute the colored fan of the normalization. As a corollary, we describe the conjugacy classes of conics in the adjoint variety and show smoothness of the normalization. Similar results on the Chow scheme of the adjoint variety are also presented.