Some remarks on smooth projective varieties of small degree and codimension

TL;DR

This paper provides simplified proofs of existing theorems on the structure of smooth projective varieties with small degree and codimension, and improves degree bounds for identifying complete intersections.

Contribution

It offers a quick proof of Ballico-Chiantini's theorem and refines Barth-Van de Ven's degree bound for complete intersections in projective varieties.

Findings

Proof of Ballico-Chiantini's theorem for Fano or Calabi-Yau varieties of dimension ≥ 4 in codimension two

Improved degree bound from ~0.63·n^{1/2} to ~0.79·n^{2/3} for complete intersections

Enhanced understanding of the relationship between degree, dimension, and codimension in projective varieties

Abstract

The purpose of this note is twofold. First, we give a quick proof of Ballico-Chiantini's theorem stating that a Fano or Calabi-Yau variety of dimension at least 4 in codimension two is a complete intersection. Second, we improve Barth-Van de Ven's result asserting that if the degree of a smooth projective variety of dimension $n$ is less than approximately $0.63 \cdot n^{1/2}$, then it is a complete intersection. We show that the degree bound can be improved to approximately $0.79 \cdot n^{2/3}$.