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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