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Minimal surface in the 4-quadric. (English) Zbl 0602.14034

The paper deals with the classification problem of smooth surfaces X in the Grassmannian of lines of \({\mathbb{P}}^ 3\). The author considers the following cases for \(X: (a)\quad rational\) \({\mathbb{P}}^ 1\)-bundle; \((b)\quad {\mathbb{P}}^ 1-bundle\) over an elliptic curve \((c)\quad \min imal\) surface of Kodaira dimension \(0.\) In cases (a) and (b) she gives a complete description of all possible cases. As to case (c) the author shows that either \((i)\quad d=\deg (X)=6\quad or\quad 8\) and X is a K 3 complete intersection, \((ii)\quad d=10\quad or\quad 12\) and X is an Enriques surface, or \((iii)\quad d=14\) and X is an abelian or hyperelliptic surface.
Reviewer: C.Turrini

MSC:

14J10 Families, moduli, classification: algebraic theory
14M15 Grassmannians, Schubert varieties, flag manifolds
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