Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Threefolds containing all curves are rationally connected

Sixuan Lou

Abstract

Any smooth projective curve embeds into $\mathbb{P}^3$. More generally, any curve embeds into a rationally connected variety of dimension at least three. We prove conversely that if every curve embeds in a threefold $X$, then $X$ is rationally connected. In particular "all curves embed" is a birational property for threefolds.

Threefolds containing all curves are rationally connected

Abstract

Any smooth projective curve embeds into . More generally, any curve embeds into a rationally connected variety of dimension at least three. We prove conversely that if every curve embeds in a threefold , then is rationally connected. In particular "all curves embed" is a birational property for threefolds.

Paper Structure

This paper contains 8 sections, 12 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Parametrizing morphisms
  5. MRC-fibration
  6. Proofs
  7. Surface and product threefold
  8. General threefold

Key Result

Theorem 1.3

Let $X$ be a smooth projective threefold where all curves embed, then $X$ is rationally connected.

Theorems & Definitions (24)

  • Theorem 1.3: \ref{['threefold']}
  • Theorem 1.4: \ref{['rat-surface']}
  • Definition 1.5
  • Corollary 1.6
  • Remark 1.7
  • Definition 2.1
  • Proposition 2.2: Proposition 1, sankaranSmoothRationallyConnected2011
  • proof : Sketch
  • Lemma 3.1
  • proof
  • ...and 14 more