On projective manifolds swept out by cubic varieties

TL;DR

This paper investigates the structure of projective manifolds covered by cubic varieties, revealing conditions for extremal contractions and providing classifications of certain manifolds based on their cubic and Segre structures.

Contribution

It introduces new classifications of projective manifolds swept out by cubic hypersurfaces and Segre threefolds, and characterizes smooth cubic hypersurfaces.

Findings

Manifolds swept out by high-dimensional cubic hypersurfaces admit linear projective bundle or cubic fibration structures.

Classification of manifolds of dimension at most five swept out by Segre threefolds.

Complete classification of five-dimensional manifolds swept out by planes.

Abstract

We study structures of embedded projective manifolds swept out by cubic varieties. We show if an embedded projective manifold is swept out by high-dimensional smooth cubic hypersurfaces, then it admits an extremal contraction which is a linear projective bundle or a cubic fibration. As an application, we give a characterization of smooth cubic hypersurfaces. We also classify embedded projective manifolds of dimension at most five swept out by copies of the Segre threefold P^1\timesP^2. In the course of the proof, we classify projective manifolds of dimension five swept out by planes.