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Scroll surfaces in $$Gr(1,{\mathbb{P}}^ 3)$$. (English) Zbl 0619.14026
Algebraic varieties of small dimension, Proc. Int. Conf., Turin/Italy 1985, Rend. Semin. Mat., Torino, Fasc. Spec. 69-75 (1986).
[For the entire collection see Zbl 0611.00006.]
A scroll surface in $${\mathbb{P}}^ 5$$ is a geometrically ruled surface whose ruling $${\mathbb{P}}^ 1$$’s are lines in $${\mathbb{P}}^ 5$$. This paper contains a geometrical description of all the scroll surfaces which are contained in the Grassmann manifold $$Gr(1,{\mathbb{P}}^ 3)\subset {\mathbb{P}}^ 5$$. The highest degree which occurs is 6; these surfaces are isomorphic to $${\mathbb{P}}({\mathcal O}_ C+L_ 0)$$, where $$L_ 0$$ is a non-trivial line bundle of degree 0 over an elliptic curve C.

##### MSC:
 14J25 Special surfaces 14M15 Grassmannians, Schubert varieties, flag manifolds 14N05 Projective techniques in algebraic geometry