Normal elliptic scrollar varieties
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In his classical memoir on the projective classification of elliptic ruled surfaces Corrado Segre described in particular two most general normal types, of even and odd order respectively, of which the former has precisely two minimum directrix curves, while the latter has an elliptic pencil of such curves. The present paper extends this work to normal elliptic scrollar varieties of dimension k, defining and describing k most general types of such varieties. Particular attention is paid to one of these types, which we call the simploid, in which the points of the variety correspond to the unordered sets of k values of an elliptic parameter (modulo its periods). The paper concludes with the identification of a series of self-dual « linked pairs » of such scrollar varieties, of which the simplest example is that of the elliptic quintic ruled surface and the elliptic quintic scrollar threefold in four-dimensional space.
KeywordsGeneral Type Normal Type Projective Classification Elliptic Parameter Elliptic Pencil
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- C. Segre,Remarques sur les transformations uniformes des coμrbes elliptiques en ellesmêmes, Opere, Vol. 1, Rome, 1957, pp. 36–55.Google Scholar
- C. Segre,Ricerche sulle rigate ellittiche di qualunque ordine, Opere, Vol 1, Rome, 1957, pp. 56–77.Google Scholar
- C. Segre,Sulle varietà algebriche composte di una serie semplicemente infinita di spazi, Opere, Vol. 1, Rome, 1957, pp. 114–118.Google Scholar
- F. Enriques — O. Chisini,Teoria geometrica delle equazioni e delle funzioni algebriche, Zanichelli, Bologna; Vol. III, 1924 and Vol. IV, 1934.Google Scholar
- J. G. Semple — L. Roth,Introduction to algebraic geometry, Oxford, 1949.Google Scholar