Geodesic
Threefolds of Order One in the Six-Quadric
Informatics and Automation, Multidimensional algebraic geometry, Tome 264 (2009), pp. 25-36.

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Consider the smooth quadric $Q_6$ in $\mathbb P^7$. The middle homology group $H_6(Q_6,\mathbb Z)$ is isomorphic to $\mathbb Z\oplus\mathbb Z$, with a basis given by two classes of linear subspaces. We classify all threefolds of bidegree $(1,p)$ inside $Q_6$. 
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L. Borisov; J. Viaclovsky. Threefolds of Order One in the Six-Quadric. Informatics and Automation, Multidimensional algebraic geometry, Tome 264 (2009), pp. 25-36. http://geodesic.mathdoc.fr/item/TRSPY_2009_264_a2/

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