Quadratic Varieties of Small Codimension

TL;DR

This paper classifies certain smooth projective varieties defined by quadratic equations, specifically those with dimension just below the extremal case, advancing the understanding of their geometric structure.

Contribution

It extends previous classifications of quadratic varieties by analyzing the case where the dimension is two less than the extremal case, filling a gap in the classification.

Findings

Classified quadratic varieties with dimension n=2c-1

Extended the Hartshorne conjecture results to new cases

Provided a detailed geometric characterization of these varieties

Abstract

Let $X \subset \mathbb{P}^{n+c}$ be a nondegenerate smooth projective variety of dimension $n$ defined by quadratic equations. For such varieties, P. Ionescu and F. Russo proved the Hartshorne conjecture on complete intersections, which states that $X$ is a complete intersection provided that $n \geq 2c+1$. As the extremal case, they also classified $X$ with $n=2c$. In this paper, we classify $X$ with $n=2c-1$.