Euler-symmetric complete intersection in projective space

TL;DR

This paper investigates Euler-symmetric projective varieties, showing that when they are complete intersections, they are formed by hyperquadrics and have a specific geometric structure related to their fundamental forms.

Contribution

It characterizes Euler-symmetric complete intersections as hyperquadric intersections and analyzes their fundamental forms at general points.

Findings

Euler-symmetric complete intersections are hyperquadric complete intersections.

The base locus of the second fundamental form is also a complete intersection.

Provides new insights into the structure of Euler-symmetric varieties.

Abstract

Euler-symmetric projective varieties, introduced by Baohua Fu and Jun-Muk Hwang in 2020, are nondegenerate projective varieties admitting many $\mathbb{C}^{\times}$-actions of Euler type. They are quasi-homogeneous and uniquely determined by their fundamental forms at a general point. In this paper, we study complete intersections in projective spaces which are Euler-symmetric. It is proven that such varieties are complete intersections of hyperquadrics and the base locus of the second fundamental form at a general point is again a complete intersection.