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Published online by Cambridge University Press: 24 October 2008
1. The work of this paper was undertaken with a view to finding out what ruled surfaces can be determined by incidences, i.e. generated by the lines which meet a certain set of spaces which I shall call a base. Such ruled surfaces I shall call incidence scrolls. In [3] the lines which meet three lines generate a quadric surface. In [4] it is easy to show that a base consisting of a line and three planes gives the general rational quartic scroll, while the lines which meet five planes in [4] give the general elliptic quintic scroll. One might be tempted to think that at least all the rational normal scrolls could be obtained as incidence scrolls by taking for base a suitable number of spaces containing directrix curves, but unfortunately there is a residual surface except in the case of the rational scrolls of general type and of those with a directrix line.
* See Segre, , Encykl. Math. Wiss., iii C 7, 818.Google Scholar
* For a fuller statement and for references, see Segre, , Encykl. Math. Wiss., iii C 7, 911, 912.Google Scholar
* By theorem (4).