Threefolds of order one in the six-quadric

TL;DR

This paper classifies all three-dimensional algebraic varieties of a specific bidegree within a smooth six-dimensional quadric, enhancing understanding of their geometric structure.

Contribution

It provides a complete classification of threefolds of bidegree (1,p) in the six-quadric Q_6, a previously unresolved problem in algebraic geometry.

Findings

Classification of threefolds of bidegree (1,p) in Q_6

Description of their geometric properties

Insights into the homology classes of these threefolds

Abstract

Consider the smooth quadric Q_6 in P^7. The middle homology group H_6(Q_6,Z) is two-dimensional with a basis given by two classes of linear subspaces. We classify all threefolds of bidegree (1,p) inside Q_6.