Manifolds covered by lines and the Hartshorne Conjecture for quadratic manifolds

TL;DR

This paper proves the Hartshorne Conjecture for quadratic manifolds by analyzing their line coverage and characterizing when they are complete intersections, providing a classification of such manifolds.

Contribution

It establishes the Hartshorne Conjecture for quadratic manifolds and characterizes when these manifolds are complete intersections based on their line varieties.

Findings

Quadratic manifolds satisfying the conjecture are classified.

Manifolds covered by lines are Fano and often complete intersections.

The variety of lines through a point determines the manifold's structure.

Abstract

Small codimensional embedded manifolds defined byequations of small degree are Fano and covered by lines. They are complete intersections exactly when the variety of lines through a general point is so and has the right codimension. This allows us to prove the Hartshorne Conjecture for manifolds defined by quadratic equations and to obtain the list of such Hartshorne manifolds.