Quadratic Varieties of Small Codimension

Abstract

Let 𝑋 ⊂ ℙⁿ⁺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 ≥ 2𝑐 + 1. As the extreme case, they also classified 𝑋 with 𝑛 = 2𝑐. In this paper, we classify 𝑋 with 𝑛 = 2𝑐 − 1.