Geometric properties of projective manifolds of small degree

TL;DR

This paper investigates the geometric properties of smooth projective varieties with small degree, classifying their structure, connectivity, and Cox rings, and establishing bounds for Calabi-Yau conditions.

Contribution

It classifies varieties of degree up to r+2, proves their simple and rational connectivity, and analyzes Cox ring finite generation with specific degree bounds.

Findings

Varieties of degree ≤ r+2 are simply connected and rationally connected except in few cases.

Cox rings are finitely generated for degree ≤ r, with counterexamples for degree r+1, r+2.

Non-uniruled varieties with degree ≤ n(r−n)+2 are Calabi-Yau, with sharp bounds.

Abstract

The aim of this paper is to study geometric properties of non-degenerate smooth projective varieties of small degree from a birational point of view. First, using the positivity property of double point divisors and the adjunction mappings, we classify smooth projective varieties in $\mathbb P^r$ of degree $d \leq r+2$, and consequently, we show that such varieties are simply connected and rationally connected except in a few cases. This is a generalization of P. Ionescu's work. We also show the finite generation of Cox rings of smooth projective varieties in $\mathbb P^r$ of degree $d \leq r$ with counterexamples for $d=r+1, r+2$. On the other hand, we prove that a non-uniruled smooth projective variety in $\mathbb P^r$ of dimension $n$ and degree $d \leq n(r-n)+2$ is Calabi-Yau, and give an example that shows this bound is also sharp.